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POSTMASTER: send address changes to Pacific Journal of Mathematics, P.O. Box 4163, Berkeley, CA 94704-0163. PUBLISHED BY PACIFIC JOURNAL OF MATHEMATICS at the University of California, Berkeley 94720-3840 A NON-PROFIT CORPORATION Typeset in LATEX Copyright ©2006 by Pacific Journal of Mathematics PACIFIC JOURNAL OF MATHEMATICS Vol. 191, No. 1, 1999 EXPLICIT CAYLEY TRIPLES IN REAL FORMS OF E7 Dragomir Ž. –D oković Let g be a noncompact real form of the simple complex Lie algebra gc of type E7 . Up to isomorphism, there are exactly three such algebras: EV, EVI, and EVII in Cartan notations. For each of these algebras we obtain a list of representatives of the adjoint orbits of standard triples (E, H, F ), i.e., triples {E, H, F } ⊂ g spanning a subalgebra isomorphic to sl2 (R), and such that [H, E] = 2E, [H, F ] = −2F , and [F, E] = H. These representative standard triples are chosen to be Cayley triples with respect to a fixed Cartan decomposition of g. 1. Introduction. The nilpotent adjoint orbits in noncompact real forms of exceptional simple Lie algebras have been classified in our papers [4, 5]. This classification is based on the so-called Kostant-Sekiguchi bijection for which we refer to [2] and [7]. See Section 5 for a more detailed discussion of this bijection. In our first two papers mentioned above, we did not compute the representatives of the nilpotent orbits. This paper is a sequel to [7] and we shall freely use the notations introduced there. In that paper we have compiled a list of representatives of G-orbits of standard triples (E, H, F ) in g where g is a noncompact real form of gc , and gc is a simple complex Lie algebra of type G2 , F4 , or E6 . In fact these representative triples were chosen to be real Cayley triples with respect to a fixed Cartan decomposition of g. In the present paper we accomplish the same objective for noncompact real forms g of gc when the latter is of the type E7 . Up to Gc -conjugacy, there are exactly 3 such real forms. They are denoted by EV, EVI, EVII or E7(7) , E7(−5) , E7(−25) , respectively. The nilpositive elements E of these representative Cayley triples are representatives of the nonzero nilpotent adjoint orbits. 1 DRAGOMIR Ž. –D OKOVIĆ 2 By using a result from our recent note [6] it is easy to determine which complex nilpotent adjoint orbits possess real points. It is more delicate to determine the number of real nilpotent orbits that are contained in a given complex orbit. We record here that D.R. King [8, p. 254] has detected an error in [5, Table VIII]. Namely the last entry for orbit 5 of that table should be sl(3, C) instead of 2su(2, 1). Several misprints in our paper [4] have been mentioned in [7]. There is one more: Namely on p. 515, in Table XII, the labels “020220 0” of the orbit No. 31 (given in the second column) should be replaced by “020220 2”. Consequently these labels should also be corrected in [2, p. 158]. 2. The root system of E7 . We denote by h a maximally split Cartan subalgebra of g which is stable under the Cartan involution θ, and by hc its complexification. The number of positive roots of gc = E7 is N = 63. The positive roots are enumerated as α1 , α2 , . . . , αN with Π = {α1 , . . . , α7 } as a base. The enumeration is chosen so that the heights increase, i.e., ht(αi ) ≤ ht(αj ) for i < j. The negative root −αi is written also as α−i . The extended Dynkin diagram of E7 is given in Fig. 1. α−63 b α1 b α3 b α4 b α5 b α6 b α7 b b α 2 Figure 1. Since E7 is simply laced, if αi = k1 α1 + · · · + k7 α7 is a positive root, then the corresponding coroot Hi is given by Hi = k1 H1 + · · · + k7 H7 . Our enumeration of positive roots αi (and coroots Hi ) is given in Table 1. EXPLICIT CAYLEY TRIPLES IN REAL FORMS OF E7 3 Table 1. Positive roots of E7 . i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 αi , H i 1000000 0100000 0010000 0001000 0000100 0000010 0000001 1010000 0101000 0011000 0001100 0000110 0000011 1011000 0111000 0101100 0011100 0001110 0000111 1111000 1011100 i 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 αi , Hi 0111100 0101110 0011110 0001111 1111100 1011110 0112100 0111110 0101111 0011111 1112100 1111110 1011111 0112110 0111111 1122100 1112110 1111111 0112210 0112111 1122110 i 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 αi , Hi 1112210 1112111 0112211 1122210 1122111 1112211 0112221 1123210 1122211 1112221 1223210 1123211 1122221 1223211 1123221 1223221 1123321 1223321 1224321 1234321 2234321 3. The structure constants of E7 . As in [7], we use an algorithm of J. Kurtzke [9] to fix the choice of a Chevalley basis of gc : Hi , 1 ≤ i ≤ 7; Xi , X−i , 1 ≤ i ≤ 63. If αi + αj = αk , then [Xi , Xj ] = N (i, j)Xk . As all roots of E7 have the same length, the nonzero structure constants are ±1, i.e., N (i, j) = ε(i, j). We specify that N (1, 3) = +1. Then, by Kurtzke’s algorithm, N (3, 4) = N (5, 6) = −1, N (4, 2) = N (4, 5) = N (6, 7) = +1, and N (i, j) = +1 whenever αi + αj is a root and 1 ≤ i ≤ 7 < j ≤ 63. Furthermore all other N (i, j)’s are uniquely determined. For the convenience of the reader, we list in the Appendix, the nonzero structure constants N (i, j) for all i > 0. For i < 0 one can use the formula N (−i, −j) = N (i, j). 4 DRAGOMIR Ž. –D OKOVIĆ 4. The conjugation σ. We recall that σ denotes the conjugation of gc with respect to g, and that hc is σ-invariant. The action of σ on hc induces naturally an action on the dual space of hc which preserves Φ. As σ acts on Φ as an automorphism, it suffices to know the action of σ on Π. If σ(αi ) = αj we also write σ(i) = j. Note that σ(i) = j implies that σ(−i) = −j. One can further assume that the Chevalley basis has been chosen so that, in addition to the properties mentioned earlier, the action of σ on the Xi ’s is given by σ(Xi ) = ξi Xσ(i) where ξi = ±1. We recall that ξi = 1 whenever αi ∈ Φ0 , and ξ−i = ξi for all i. For all three noncompact real forms g of gc we may choose ξi = 1 for 1 ≤ i ≤ 7. With this information, one can compute the coefficients ξi for arbitrary i. For reader’s convenience we list the vectors σ(Xi ), i > 0, in Table 2 for EVI and Table 3 for EVII. When g is of type EV, i.e., g is the split real form of gc , then the action of σ on Φ is trivial. Hence in that case we have σ(Xi ) = Xi for all i. Table 2. EVI = E7(−5) : Action of σ on the Xi ’s. i σ(Xi ) 1 X1 2 X−2 3 X3 4 X16 5 X−5 6 X19 7 X−7 8 X8 9 X11 10 −X22 11 X9 12 X13 13 X12 14 −X26 15 X17 16 X4 17 X15 18 X30 19 X6 20 X21 21 X20 i 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 σ(Xi ) −X10 −X25 X36 −X23 −X14 X39 X28 −X31 X18 −X29 X32 −X34 −X33 −X45 X24 X37 −X48 X27 X41 X40 −X51 i 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 σ(Xi ) X44 X43 −X35 X47 X46 −X38 X49 X56 −X42 X52 −X54 −X53 X55 X50 −X60 X59 X58 −X57 X61 X62 X63 EXPLICIT CAYLEY TRIPLES IN REAL FORMS OF E7 Table 3. EVII = E7(−25) : Action of i σ(Xi ) i σ(Xi ) 1 X37 22 X−22 2 X−2 23 X24 3 X−3 24 X23 4 X−4 25 −X36 5 X−5 26 −X14 6 X40 27 X53 7 X7 28 X−28 8 X32 29 −X18 9 X−9 30 X31 10 X−10 31 X30 11 X−11 32 X8 12 X35 33 −X50 13 −X45 34 X56 14 −X26 35 X12 15 X−15 36 −X25 16 X−16 37 X1 17 X−17 38 X46 18 −X29 39 −X54 19 X41 40 X6 20 X21 41 X19 21 X20 42 −X43 5 σ on the Xi ’s. i σ(Xi ) 43 −X42 44 X51 45 −X13 46 X38 47 −X48 48 −X47 49 X49 50 −X33 51 X44 52 −X62 53 X27 54 −X39 55 X61 56 X34 57 −X60 58 X59 59 X58 60 −X57 61 X55 62 −X52 63 X63 5. The Kostant-Sekiguchi bijection. We refer the reader to our previous paper [7] for definition of G, K, Gc , K c , standard triples, normal triples, real and complex Cayley triples, etc. In the following diagram we exhibit several important sets on which some of the above groups act and some natural maps between these sets Set Group Nonzero nilpotent elements in g ↑α Standard triples in g ↑β Real Cayley triples in g ↓γ Complex Cayley triples in gc ↓δ Normal triples in gc ↓ε Nonzero nilpotent elements in pc G G K K Kc Kc 6 DRAGOMIR Ž. –D OKOVIĆ The map α (resp. ε) sends the standard (resp. normal) triple (E, H, F ) to its nilpositive part E. The maps β and δ are the inclusion maps. The map γ is the Cayley transformation. Clearly γ is bijective and K-equivariant. The maps β and δ are also K-equivariant. The map α is G-equivariant while ε is K c -equivariant. We shall prove below that, on the level of orbits, each of these maps induces a bijection. Since α is G-equivariant, it induces a map α∗ from the set of G-orbits of standard triples in g to the set of G-orbits of nonzero nilpotent elements of g. One defines similarly the maps β ∗ , γ ∗ , δ ∗ , and ε∗ . Proposition 1. Each of the maps α∗ , . . . , ε∗ is a bijection. Proof. All the references in this proof are to the book [2]. Since γ is bijective, so is γ ∗ . The map α∗ is surjective by Theorem 9.2.1, and injective by Theorem 9.2.3. The map ε∗ is surjective by Theorem 9.4.2, and injective by Theorem 9.4.3. The map β ∗ is surjective by Theorem 9.4.1. The fact that δ ∗ is surjective is shown in the proof of Theorem 9.5.1. We shall prove now that β ∗ is injective. Thus, if (E, H, F ) and (E 0 , H 0 , F 0 ) are real Cayley triples which are G-conjugate, we have to show that they are also K-conjugate. As H, H 0 ∈ p, Lemma 9.4.5 shows that H and H 0 are K-conjugate. Hence without any loss of generality we may assume that H 0 = H. By our hypothesis, there exists g ∈ G such that g · (E, H, F ) = (E 0 , H, F 0 ). In particular, g ∈ ZG (H) and g · (E + F ) = E 0 + F 0 . Since H is semisimple, the centralizer ZG (H) is reductive. By applying the proof of Lemma 9.4.5 and observing that E + F, E 0 + F 0 ∈ k, we conclude that there exists k ∈ ZK (H) such that k · (E + F ) = E 0 + F 0 . The formula [H, E + F ] = 2(E − F ) now implies that k · (E − F ) = E 0 − F 0 . It follows that k · E = E 0 and k · F = F 0 , i.e., the real Cayley triples (E, H, F ) and (E 0 , H, F 0 ) are K-conjugate. By Theorem 9.5.1 the composite map ε∗ ◦ δ ∗ ◦ γ ∗ ◦ (β ∗ )−1 ◦ (α∗ )−1 is a bijection. It follows that δ ∗ must be also injective. The composite map α∗ ◦ β ∗ ◦ (γ ∗ )−1 ◦ (δ ∗ )−1 ◦ (ε∗ )−1 is the Kostant– Sekiguchi bijection from the set of nonzero nilpotent K c -orbits in pc to the set of nonzero nilpotent G-orbits in g. Explicitly, if (E, H, F ) is a real Cayley triple and (E 0 , H 0 , F 0 ) its Cayley transform, then the orbit G · E corresponds to the orbit K c · E 0 . Define a partial order in the set of nilpotent K c -orbits in pc by setting O1 ≥ O2 if O2 is contained in the closure of the orbit O1 . Define similarly the partial order in the set of nilpotent G-orbits in g. It was shown very recently [1] that the Kostant–Sekiguchi bijection preserves these partial orders. 6. Two invariants. EXPLICIT CAYLEY TRIPLES IN REAL FORMS OF E7 7 Let (E, H, F ) be a real Cayley triple and (E 0 , H 0 , F 0 ) its Cayley transform. In order to distinguish between various G-orbits in g which are contained in the same nonzero nilpotent Gc -orbit in gc , we use two invariants: tr := trace ad(H 0 )2 |kc and inv := dim Zkc (H 0 ). The second one was used in our previous paper [7], while the first one is easier to compute. It is evident from our tables that in some instances tr fails to distinguish two orbits and we have to use inv. In a few instances inv fails, while tr succeeds to distinguish two orbits. Our method for computing the representative real Cayley triples (E, H, F ) in g is described in detail in [7]. In a relatively few cases the method fails and we had to do extensive computations to find the desired representatives. We shall describe the difficulties that arise on one such example in Section 9 (the most difficult case). 7. Pairs of orbits G · E and −G · E. If E 6= 0 is a nilpotent element in g, then there exists an automorphism of g which maps E to −E (see [3]). In general, this automorphism is not inner and so E and −E may belong to different G-orbits. One can decide whether or not G · E = −G · E by means of the following proposition. Proposition 2. Let (E, H, F ) be a real Cayley triple and (E 0 , H 0 , F 0 ) its Cayley transform. Then G · E = −G · E if and only if K c · H 0 = −K c · H 0 . Proof. The triple (−E, H, −F ) is another real Cayley triple, and its Cayley transform is (−F 0 , −H 0 , −E 0 ). We have G · E = −G · E if and only if the real Cayley triples (E, H, F ) and (−E, H, −F ) are G-conjugate. By the properties of the Sekiguchi bijection, this is the case if and only if the corresponding complex Cayley triples, namely (E 0 , H 0 , F 0 ) and (−F 0 , −H 0 , −E 0 ) are K c -conjugate. The latter condition is equivalent to the K c -conjugacy of H 0 and −H 0 . The characteristics H 0 of the nonzero nilpotent K c -orbits in pc are known [4]. Since H 0 ∈ hc , H 0 and −H 0 are K c -conjugate if and only if they belong to the same orbit of the Weyl group W (kc , gc ). Hence it is easy to check whether or not G · E = −G · E. When g is of Cartan type EVI, then Aut (g) = G, and so G · E = −G · E for all nonzero nilpotent elements E ∈ g. In the other two cases, when g is of Cartan type EV or EVII, there exist nonzero nilpotent elements E ∈ g such that G · E 6= −G · E. Such pairs of orbits are easily recognizable from Tables 4 and 6 because we give their representatives jointly as ±E. 8 DRAGOMIR Ž. –D OKOVIĆ 8. Tables of real Cayley triples. We give here lists of representatives (E, H, F ) for K-orbits of real Cayley triples in g, a noncompact real form of the simple complex Lie algebra gc of type E7 . We record only the elements E and H because F can be easily computed by using F = θ(E) = θσ(E) = σu (E) and σu (Xi ) = X−i (for all i). For the neutral element H, we list both the labels αi (H) for 1 ≤ i ≤ 7 and the coefficients ki in the linear combination H = k1 H1 + · · · + k7 H7 . The nilpositive element E is written explicitly as a linear combination of the root vectors Xi ,i > 0. Note that the element H is always the characteristic of the nilpotent orbit Gc · E. On the other hand there is no natural choice for the nilpositive element E. Our preference was to choose E so that its support (i.e. the number of nonzero coefficients) is minimal even though this may have a drawback of introducing irrational coefficients. EXPLICIT CAYLEY TRIPLES IN REAL FORMS OF E7 9 Table 4. Cayley triples in EV = E7(7) . αi (H) 1 1000000 2 0000010 3, 4 0000002 5 0010000 6 2000000 7 2000000 8, 9 0100001 10, 11 1000010 12 ki 2, 2, 3, 4, 3, 2, 1 2, 3, 4, 6, 5, 4, 2 2, 3, 4, 6, 5, 4, 3 3, 4, 6, 8, 6, 4, 2 4, 4, 6, 8, 6, 4, 2 4, 4, 6, 8, 6, 4, 2 3, 5, 6, 9, 7, 5, 3 4, 5, 7, 10, 8, 6, 3 1000010 4, 5, 7, 10, 8, 6, 3 13, 14 0001000 4, 6, 8, 12, 9, 6, 3 15 0001000 4, 6, 8, 12, 9, 6, 3 16, 17 0200000 4, 7, 8, 12, 9, 6, 3 18, 19 0200000 4, 7, 8, 12, 9, 6, 3 20 2000010 6, 7, 10, 14, 11, 8, 4 21 0000020 4, 6, 8, 12, 10, 8, 4 22, 23 2000002 6, 7, 10, 14, 11, 8, 5 24 0010010 5, 7, 10, 14, 11, 8, 4 25 26 1001000 6, 8, 11, 16, 12, 8, 4 0020000 6, 8, 12, 16, 12, 8, 4 27 0020000 6, 8, 12, 16, 12, 8, 4 28, 29 1000101 6, 8, 11, 16, 13, 9, 5 30 2020000 10, 12, 18, 24, 18, 12, 6 31, 32 0110001 6, 9, 12, 17, 13, 9, 5 33, 34 0110001 6, 9, 12, 17, 13, 9, 5 35, 36 0001010 6, 9, 12, 18, 14, 10, 5 E X63 X49 + X63 ±(X7 + X49 + X63 ) X37 + X55 + X61 X1 +√X37 + X55 + X61 2(X1 + X62 ) ±(X30 + X47 + X53 + X59 ) ±(X27 + X39 + X49 +X53 − X54 ) X27 + X39 + X49 +X53 + X54 ±(X28 + X38 + X47 −X48 + X49 ) X28 + X38 + X46 +X47 + X48 + X49 ±(X2 + X28 − X38 + X46 +X47 + X48 + X49 ) ±(X2 + X28 + X38 + X46 √+X47 + X48 + X49 ) √ 3(X1 + X37 ) + 2X49 2(X6 + X19 + X50 + X56 ) ±[X7 + 2X49 √ + 3(X √ 1 + X37 )] X37 + 2(X24 + X33 +X41√+ X48 ) X28 + 2X49 + 3(X14 + X26 ) X3√− X28 + 2X49 + 3(X14 + X26 ) X3√+ X28 + 2X49 + 3(X14 + X26 ) ±[X√19 + X40 + 2X41 + √ 3(X21 + √X33 )] 10X1 + 6(X3 +X28 + X49 ) ±[X15 √ + X30 + 2X31 − X40 + 3(X26 + X38 )] ±[X15 √ + X30 + 2X31 + X40 + 3(X26 + X38 )] ±[−X18 + √ X30 + 2X32 + X33 +X34 + 3(X29 − X31 )] tr inv 32 31 64 23 96 39 96 19 128 37 128 31 128 19 160 21 160 15 192 25 192 13 224 49 224 25 320 256 352 15 23 23 288 11 352 384 11 19 384 21 384 11 896 19 416 11 416 13 448 17 DRAGOMIR Ž. –D OKOVIĆ 10 Table 4. (continued) 37 38 39, 40 41, 42 43 44, 45 46, 47 48, 49 50 51, 52 53 54 55, 56 57, 58 59 60, 61 62 αi (H) ki √ E 0001010 6, 9, 12, 18, 2X37√ + 2(X29 + X34 ) 14, 10, 5 + 3(X18 + X30 ) 2000020 8, 10, 14, 20, 2(X1 + X37 ) √ 16, 12, 6 + 3(X6 + X19 + X40 + X41 ) 0000200 6, 9, 12, 18, ±[X5 + 2X18√+ X28 + X29 15, 10, 5 −X30 − X31 + 3(X26 + X47 )] 0000200 6, 9, 12, 18, ±[X30 + 2X √ √ 47 15, 10, 5 + 2(X28 + X33 ) + 3(X5 + X18 )] 2000020 8, 10, 14, 20, 2(X1 + X37 ) √ 16, 12, 6 + 3(−X6 + X √19 + X40 + X41 ) 2000022 10, 13, 18, ±[3X7√+ 2 2(X6 + X40 ) 26, 21, 16, 9 + 5(X1 + √ X37 )] 2110001 10, 13, 18, ±[X + 10X1 30 √ 25, 19, 13, 7 + 6(X15 + X31 + X40 )] 1001010 8, 11, 15, 22, √ ±[X28 + 2(X14 + X26 ) 17, 12, 6 + 3(X18 + X29 + X30 − X31 )] 1001010 8, 11, 15, 22, X28√+ 2(X14 + X26 ) 17, 12, 6 + 6(X18 + X √36 ) 2001010 10, 13, 18, ±[X − X + 10X1 18 30 √ 26, 20, 14, 7 + 6(X28 + X29√+ X31 )] 2001010 10, 13, 18, X √18 − X30 + 10X1 26, 20, 14, 7 + 6(X28 +√ X29 + X31 ) 0002000 8, 12, 16, 24, 2(X4 + X√16 ) + 2(X15 + X21 ) 18, 12, 6 + 6(X24 + X39√ ) 2000200 10, 13, 18, ±[X − X + X + 10X1 5 18 30 √ 26, 21, 14, 7 6(X28 + X29 + X√ 31 )] 2000200 10, 13, 18, ±[X + X + X + 10X1 18 30 √5 26, 21, 14, 7 6(X28 + X + X 31 )] √ 29 1001020 10, 14, 19, 3X28√ + 2 2(X6 + X19 ) 28, 22, 16, 8 + 5(X14√ + X26 ) 1001012 10, 14, 19, ±[3X7 + X + 2 2(X18 + X29 ) 28 √ 28, 22, 16, 9 +√ 2(X18 + X29 ) + 5(X14√+ X26 )] 0020020 10, 14, 20, X3 − 3X√28 + 2 2(X6 + X19 ) 28, 22, 16, 8 + 5(X14 + X26 ) tr inv 448 9 640 19 480 33 480 17 640 15 1120 15 928 11 672 11 672 7 960 13 960 7 768 13 992 25 992 13 1120 7 1152 7 1152 11 EXPLICIT CAYLEY TRIPLES IN REAL FORMS OF E7 11 Table 4. (continued) αi (H) ki 0020020 10, 14, 20, 28, 22, 16, 8 64, 65 0110102 10, 15, 20, 29, 23, 16, 9 63 66 2020020 14, 18, 26, 36, 28, 20, 10 67, 68 0002002 10, 15, 20, 30, 23, 16, 9 69, 70 0002002 10, 15, 20, 30, 23, 16, 9 71 0002020 12, 18, 24, 36, 28, 20, 10 72, 73 2110102 14, 19, 26, 37, 29, 20, 11 74, 75 2110110 14, 19, 26, 37, 29, 20, 10 76, 77 2002002 14, 19, 26, 38, 29, 20, 11 78, 79 2002002 14, 19, 26, 38, 29, 20, 11 80 2002020 16, 22, 30, 44, 34, 24, 12 81 2002020 16, 22, 30, 44, 34, 24, 12 82, 83 2110122 18, 25, 34, 49, 39, 28, 15 E√ X3 + 3X√28 + 2 2(X6 + X19 ) + 5(X14 + X26 ) ±[X13 −√X15 − 3X23 + X24 +2 √ 2(X17 − X7 ) + √ 5(X20 + X27 )] 2(2X√ 3 + 3X28 ) √ + 14X1 +√ 10(X6 + X19 ) ±[3X7√+ 5(X4 + X16 ) + 2(X15 + X21 +2X24 + 2X√ 33 )] ±[2X + X − 10X17 27 √ 4 + 6(X √ 13 + X20 − X23 ) +√3(X26 − X19 )] 2 √3(X15 + X21 ) +√ 6(X4 + X16 ) + 10(X√6 + X19 ) ±[X13 √ + X23 + 2(2X15 √ + 3X24 ) + 10(X16√− X7 ) + 14X1 ] ±[X √ 16 + 2(3X15 +√2X17 ) + 10(X12 + X25 ) + √14X1 ] ±[X10 + √X13 + X29 + 14X8 +√ 2(3X9 + 2X18 ) + 10(X17√− X19 )] ±[X4 + √X13 + 14X1 + X23 + √2(2X15 + 3X24 ) + 10(X √ 7 − X16 )] 4X1 + 2 √ 3(X6 + X19 ) +√ 7(X4 + X16 ) + 15(X √ 15 + X17 ) 4X1 +√2 3(X12 + X13 ) +√ 7(X4 + X16 ) + 15(X √ √10 + X22 )√ ±[3√ 2X1 + 2√ 6X17 + 2 7X6 + 10X15 + 15(X7 + X16 )] tr inv 1152 13 1216 11 1920 11 1248 11 1248 21 1792 9 1984 9 1952 7 2016 9 2016 17 2688 9 2688 7 3520 7 DRAGOMIR Ž. –D OKOVIĆ 12 Table 4. (continued) αi (H) ki 2022020 22, 30, 42, 60, 46, 32, 16 85, 86 2002022 18, 25, 34, 50, 39, 28, 15 84 87, 88 2002022 18, 25, 34, 50, 39, 28, 15 89, 90 2220202 22, 31, 42, 60, 47, 32, 17 91, 92 2220222 26, 37, 50, 72, 57, 40, 21 93, 94 2222222 34, 49, 66, 96, 75, 52, 27 E √ 4(X + X 22X 19 ) + √ 6 √ 1 + 30(X4 + X ) + 16 √ √42X3 ±[X11√ + 3 2X1√+ 2 6X10 +2 √7X12 + 10X22 + 15(X √ 7 + X9 )]√ ±[−X11√+ 3 2X√ 1 + 2 6X10 +2 √7X12 + 10X22 + 15(X7 + X9 )] ±[−X2 − 4X5 √ + 4X7 +X + X + 22X 13 √ 12 √ 1 + 30(X + X ) + 9 18 √ √ 42X3 ] ±[ √26X1 + 5√ 2X3 +2 10X6 + 21X7 √ 1 √ + √ ( 33X5 − 77X2 5 √ √ −6 √ 3X9 − 6 7X√11 )] ±[√34X1 +√7X2 + 66X3 +4 6X4 + √ 3(5X5 + 3X7 ) +2 13X6 ] tr inv 4992 7 3552 7 3552 13 5088 11 7392 9 12768 7 EXPLICIT CAYLEY TRIPLES IN REAL FORMS OF E7 13 Table 5. Cayley triples in EVI = E7(−5) . 1 2 3 4 5 6 7 8 9 10 αi (H) 1000000 0000010 0000010 0010000 0010000 2000000 2000000 2000000 1000010 0001000 ki 2, 2, 3, 4, 3, 2, 1 2, 3, 4, 6, 5, 4, 2 2, 3, 4, 6, 5, 4, 2 3, 4, 6, 8, 6, 4, 2 3, 4, 6, 8, 6, 4, 2 4, 4, 6, 8, 6, 4, 2 4, 4, 6, 8, 6, 4, 2 4, 4, 6, 8, 6, 4, 2 4, 5, 7, 10, 8, 6, 3 4, 6, 8, 12, 9, 6, 3 11 0001000 4, 6, 8, 12, 9, 6, 3 E X63 X49 − X63 X49 + X63 X37 − X55 + X61 X37 + X55 + X61 X1 − X37 + X55 − X61 X1 + X37 + X55 − X61 X1 + X37 + X55 + X61 X27 + X39 + X49 + X53 − X54 X28 + X38 + X46 +X47 − X48 + X49 −X28 + X38 + X46 √+X47 − X48 + X49 √3(X1 − X37 ) + 2X49 3(X1 + X37 ) + 2X49 1 √ [X6 + X12 + X13 + X19 2 +X35 + X40 + X41 − X45 +i(X38 + X42 + X43 − X44 −X46 + X47 + X48 + X51 )] √ √ 2(X6 + X19 + X50 + X56 ) 2(X24 + X36 + X38 − X48 ) +X √37 X28 + 2X49 + √3(X14 − X26 ) X28 − 2X49 + 3(X14 − X26 ) −X√3 + X28 + 2X49 + 3(X14 − X26 ) X3√− X28 + 2X49 + 3(X14 − X26 ) X3√+ X28 + 2X49 + √ 3(X14 − X26 ) √ 10X1 + 6(X3 − X28 + X49 ) tr inv 32 37 48 33 80 25 48 37 112 21 32 67 128 37 160 35 192 19 240 19 208 15 272 368 21 21 320 49 320 352 25 13 432 368 480 21 13 39 352 23 448 21 22 2020000 10, 12, 18, 24, 608 18, 12, 6 √ √ 23 2020000 10, 12, 18, 24, 10X1 + 6(X3 + X28 + X49 ) 992 18, 12, 6 X29 − X31√+ 2X37 + iX27 528 24 0001010 6, 9, 12, 18, 14, 10, 5 −X39 + 3(X18 + X30 ) 37 12 2000010 6, 7, 10, 14, 11, 8, 4 13 2000010 6, 7, 10, 14, 11, 8, 4 14 0000020 4, 6, 8, 12, 10, 8, 4 15 0000020 4, 6, 8, 12, 10, 8, 4 16 0010010 5, 7, 10, 14, 11, 8, 4 17 1001000 6, 8, 11, 16, 12, 8, 4 18 1001000 6, 8, 11, 16, 12, 8, 4 19 0020000 6, 8, 12, 16, 12, 8, 4 20 0020000 6, 8, 12, 16, 12, 8, 4 21 0020000 6, 8, 12, 16, 12, 8, 4 21 11 14 DRAGOMIR Ž. –D OKOVIĆ Table 5. (continued) αi (H) ki E 25 2000020 8, 10, 14, 20, 2(X − X37 ) 1 √ 16, 12, 6 + 3(X6 + X19 + X40 + X41 ) 26 2000020 8, 10, 14, 20, 2(X1 + X37 ) √ 16, 12, 6 + 3(X6 + X19 + X40 + X41 ) 27 1001010 8, 11, 15, 22, √ X28 + 2(X14 − X26 ) 17, 12, 6 + 3(X √ 18 + X29 + X30 − X31 ) 28 2001010 10, 13, 18, 26, √10X1 + X18 + X30 20, 14, 7 + 6(X28 + X29 − X31 ) 29 0002000 8, 12, 16, 24, X4 + X15 + X16 + X17 18, 12, 6 +X √ 18 + X29 + X30 − X31 + 2(X9 + X10 + X11 − X22 ) +2i(X √27 − X39 ) 30 1001020 10, 14, 19, 28, 3X28√ + 2 2(X6 + X19 ) 22, 16, 8 + 5(X √ 14 − X26 ) 31 0020020 10, 14, 20, 28, X3 + 2 √2(X6 + X19 ) 22, 16, 8 +3X28 + √5(X14 − X26 ) 32 0020020 10, 14, 20, 28, −X3 + 2√2(X6 + X19 ) 22, 16, 8 +3X 28 +√ 5(X14 − X26 ) √ 33 2020020 14, 18, 26, 36, 14X1√+ 2(2X3 + 3X28 ) 28, 20, 10 √ + 10(X √ 6 + X19 ) 34 2020020 14, 18, 26, 36, 14X1√+ 2(3X28 − 2X3 ) 28, 20, 10 √+ 10(X6 + X19 ) 35 0002020 12, 18, 24, 36, 6(X4 + X15 + X16 28, 20, 10 +X17 √ + iX20 − iX21 ) + 10(X √ 6 + X19 ) 36 2002020 16, 22, 30, 44, 4X1 + 2 √ 3(X6 + X19 ) 34, 24, 12 +√ 7(X4 + X16 ) + 15(X15 + √ X17 ) 37 2022020 22, 30, 42, 60, 4(X + X ) + 22X1 19 √ 6 √ 46, 32, 16 + 42X3 + 30(X4 + X16 ) tr inv 672 33 800 17 832 9 1072 11 960 27 1328 13 1408 13 1312 23 2272 13 1888 21 2240 19 3232 15 5728 13 EXPLICIT CAYLEY TRIPLES IN REAL FORMS OF E7 15 Table 6. Cayley triples in EVII = E7(−25) . 1, 2 3, 4 5 6, 7 8, 9 10 11, 12 αi (H) 1000000 0000010 0000010 0000002 0000002 2000000 1000010 13, 14 2000010 15 0000020 16, 19 2000002 17, 18 2000020 20 2000020 21, 22 2000022 ki E tr inv 2, 2, 3, 4, 3, 2, 1 ±X63 32 47 2, 3, 4, 6, 5, 4, 2 ±(X49 − X63 ) 32 47 2, 3, 4, 6, 5, 4, 2 X49 + X63 96 31 2, 3, 4, 6, 5, 4, 3 ±(X7 − X49 − X63 ) 0 79 2, 3, 4, 6, 5, 4, 3 ±(X7 + X49 + X63 ) 128 47 4, 4, 6, 8, 6, 4, 2 X1 + X37 + X55 + X61 192 37 4, 5, 7, 10, 8, ±(X27 + X39 + X49 224 21 6, 3 +X53 − X ) 54 √ 6, 7, 10, 14, ±[2X49 + 3(X1 + X37 )] 416 31 11, 8, 4 √ 4, 6, 8, 12, 2(X6 + X34 + X40 + X56 ) 384 31 10, 8, 4 √ 6, 7, 10, 14, ±[−X7 + 2X49 + 3(X1 + X37 )] 384 31 11, 8, 5 √ 6, 7, 10, 14, ±[X7 + 2X49 + 3(X1 + X37 )] 512 47 11, 8, 5 8, 10, 14, 20, 2(X1 + X37 ) 960 19 √ 16, 12, 6 + 3(X6 √ + X19 + X40 + X41 ) 10, 13, 18, 26, 1536 31 √±[ 5(X1 + X37 ) 21, 16, 9 +2 2(X6 + X40 ) + 3X7 ] 9. An elaborate example. Let g be of type E7(−5) and Oc the complex nilpotent orbit of gc with characteristic 0000020, i.e., H = 2H6 . Then Oc ∩ g is the union of two G-orbits, namely orbits No. 14 and 15 in Table 5. The characteristic H defines a gradation of gc (and g) such that gc = 2 M g(2k)c . k=−2 g(0)c The subspace is a reductive subalgebra with 1-dimensional center and the derived subalgebra of type A1 + D5 . The derived subgroup of G(0)c is isomorphic to (Spin10 × SL2 )/Z2 . The dimensions of the spaces g(2k)c are as follows: dim g(0)c = 49 , dim g(−2)c = dim g(2)c = 32 , dim g(−4)c = dim g(4)c = 10 . 16 DRAGOMIR Ž. –D OKOVIĆ The subspace g(2)c has a basis consisting of the root vectors Xk , k ∈ I, where I = I1 ∪ I2 and I1 = {6, 12, 18, 23, 24, 27, 29, 33, 35, 38, 40, 42, 43, 46, 50, 53}, I2 = {13, 19, 25, 30, 31, 34, 36, 39, 41, 44, 45, 47, 48, 51, 54, 56}. As a G(0)c -module, g(2)c is the tensor product of a half-spin module of Spin10 and the 2-dimensional simple module of SL2 . The subspace of g(2)c spanned by the vectors Xk with k ∈ I1 (or k ∈ I2 ) is a half-spin module of Spin10 . We find several subsets J ⊂ I such that the subspace of g(2)c spanned by Xk with k ∈ J is σ-stable, for k, j ∈ J the difference αk − αj is not a root, and H belongs to the subspace spanned by all coroots Hk with k ∈ J. All of them have size 4. For instance, the set J = {6, 19, 50, 56} satisfies all the conditions mentioned above. As σ(X6 ) = X19 and σ(X50 ) = X56 (see Table 2), the vector E = (aX6 + āX19 ) + (bX50 + b̄X56 ) is real in the sense that σ(E) = E, i.e., E ∈ g(2). The equation [F, E] = H, where F = θ(E) = σu (E), implies that |a|2 = 2 |b| = 2. All possible choices for a and b produce an element E belonging to the orbit 15. This is established by computing the invariant inv which turns out to be 25 in all cases. The other sets J also produce only representatives for the orbit 15. In order to find a representative for the orbit 14 we had to undertake the following tedious calculation. Since E ∈ g(2), i.e., E ∈ g(2)c and σ(E) = E, the representative E must have the form E = (aX6 + āX19 ) + (bX12 + b̄X13 ) + (cX18 + c̄X30 ) ¯ 25 ) + (eX24 + ēX36 ) + (f X27 + f¯X39 ) + (dX23 − dX + (gX29 − ḡX31 ) + (hX33 − h̄X34 ) + (αX35 − ᾱX45 ) + (βX38 − β̄X48 ) + (γX40 + γ̄X41 ) + (δX42 − δ̄X51 ) + (εX43 + ε̄X44 ) + (ζX46 + ζ̄X47 ) + (ηX50 + η̄X56 ) + (θX53 − θ̄X54 ) where a, b, . . . , θ are some complex numbers. Since F = θ(E) = σu (E), we must have F = (āX−6 + aX−19 ) + (b̄X−12 + bX−13 ) + (c̄X−18 + cX−30 ) ¯ −23 − dX−25 ) + (ēX−24 + eX−36 ) + (f¯X−27 + f X−39 ) + (dX + (ḡX−29 − gX−31 ) + (h̄X−33 − hX−34 ) + (ᾱX−35 − αX−45 ) + (β̄X−38 − βX−48 ) + (γ̄X−40 + γX−41 ) + (δ̄X−42 − δX−51 ) + (ε̄X−43 + εX−44 ) + (ζ̄X−46 + ζX−47 ) + (η̄X−50 + ηX−56 ) + (θ̄X−53 − θX−54 ). EXPLICIT CAYLEY TRIPLES IN REAL FORMS OF E7 17 Next we have computed the bracket [F, E] by using the above expressions for E and F and the list of the structure constants given in the Appendix. As [F, E] = H where H = 4H1 + 6H2 + 8H3 + 12H4 + 10H5 + 8H6 + 4H7 , we obtain the following system of equations. Re (ef¯ + g h̄ + αβ̄ + γ ε̄) = 0 Re (cē + dḡ + β δ̄ + εζ̄) = 0 Re (cf¯ + dh̄ − αδ̄ − γ ζ̄) = 0 Re (−aᾱ + bγ̄ + f η̄ + hθ̄) = 0 Re (aβ̄ − bε̄ + eη̄ + g θ̄) = 0 Re (−aδ̄ + bζ̄ + cη̄ + dθ̄) = 0 b̄c − ad¯ + ḡα + h̄β + eγ̄ + f ε̄ + ζ̄η + δ θ̄ ac̄ + b̄d − ēα − f¯β + gγ̄ + hε̄ + ζ̄θ − δ η̄ ¯ − f ζ̄ − h̄δ + β θ̄ + ε̄η aḡ − b̄e + cγ̄ + dα ¯ + eζ̄ + ḡδ − αθ̄ − γ̄η ah̄ − b̄f + cε̄ + dβ aē + b̄g + c̄α − dγ̄ − f¯δ + hζ̄ + β η̄ − ε̄θ af¯ + b̄h + c̄β − dε̄ + ēδ − g ζ̄ − αη̄ + γ̄θ =0 =0 =0 =0 =0 =0 |f |2 + |h|2 + |β|2 + |δ|2 + |ε|2 + |ζ|2 + |θ|2 + |η|2 = 2 |a|2 + |b|2 + |c|2 + |d|2 + |e|2 + |g|2 + |α|2 + |γ|2 = 2 |e|2 + |g|2 + |α|2 + |γ|2 + |δ|2 + |ζ|2 + |θ|2 + |η|2 = 2 |c|2 + |d|2 + |e|2 + |g|2 + |θ|2 + |η|2 + 2|α|2 + 2|γ|2 = 2 + |f |2 + |h|2 . After some judicious specializations and experimentations using the Maple’s solve routine, we have found the following solution of the above system: √ a = b = 1/ √2 , c = d = e = f =√g = h = 0 , α = γ =√1/ 2 , β = δ = ε = i/ 2 , ζ = −i/ 2 , η = θ = 0 . This means that we have a real Cayley triple (E, H, F ) with 1 E = √ [X6 + X12 + X13 + X19 + X35 + X40 + X41 − X45 2 +i(X38 + X42 + X43 − X44 − X46 + X47 + X48 + X51 )]. For this triple we find that inv = 49, and so E is a representative of the orbit 14. 10. Appendix. We list here the nonzero structure constants N (i, j) of E7 for i > 0 and j arbitrary. The i-th entry in this list contains two sequences separated by a semicolon. The first (resp., second) sequence consists of those j for which N (i, j) is +1 (resp., −1). Note that each of these sequences has length 16. DRAGOMIR Ž. –D OKOVIĆ 18 The nonzero structure constants of E7 1 3,10,15,17,22,24,28,29,31,35,36,40,41,45,49,62; -8,-14,-20,-21,-26,-27,-32,-33,-34,-38,-39,-43,-44,-48,-52,-63 2 10,11,14,17,18,21,24,25,27,31,34,50,54,57,59,-9; 4,-15,-16,-20,-22,-23,-26,-29,-30,-33,-36,-39,-53,-56,-58,-60 3 9,11,16,18,23,25,30,32,38,43,44,48,52,61,-8,-10; 1,4,-15,-17,-22,-24,-29,-31,-36,-37,-42,-46,-47,-51,-55,-62 4 2,3,5,8,12,19,22,26,29,33,36,39,46,51,55,60; -9,-10,-11,-14,-18,-25,-28,-32,-35,-38,-41,-44,-50,-54,-57,-61 5 9,10,13,14,15,20,35,38,41,42,44,47,57,58,-11,-12; 4,6,-16,-17,-19,-21,-22,-26,-40,-43,-45,-46,-48,-51,-59,-60 6 5,7,11,16,17,21,22,26,28,32,37,45,48,51,54,56; -12,-13,-18,-23,-24,-27,-29,-33,-35,-38,-42,-49,-52,-55,-57,-58 7 12,18,23,24,27,29,33,35,38,40,42,43,46,50,53,-13; 6,-19,-25,-30,-31,-34,-36,-39,-41,-44,-45,-47,-48,-51,-54,-56 8 9,11,16,18,23,25,30,61,-1,-14,-37,-42,-46,-47,-51,-55; 4,28,35,40,41,45,49,-3,-20,-21,-26,-27,-33,-34,-39,-63 9 17,21,24,27,31,34,59,-4,-15,-16,-20,-23,-30,-53,-56,-58; 3,5,8,12,19,46,51,55,-2,-28,-32,-35,-38,-41,-44,-61 10 16,23,30,43,48,52,-4,-14,-15,-17,-24,-31,-37,-42,-47,-62; 1,2,5,12,19,26,33,39,60,-3,-28,-35,-41,-50,-54,-57 11 13,15,20,42,47,58,-4,-16,-17,-18,-21,-40,-43,-45,-48,-59; 2,3,6,8,29,33,36,39,55,-5,-25,-28,-32,-50,-54,-61 12 9,10,14,15,20,41,44,47,-6,-18,-19,-40,-43,-46,-59,-60; 4,7,28,32,37,54,56,-5,-23,-24,-27,-29,-33,-49,-52,-55 EXPLICIT CAYLEY TRIPLES IN REAL FORMS OF E7 13 40,43,46,50,53,-6,-19,-25,-30,-31,-34,-36,-39,-41,-44,-47; 5,11,16,17,21,22,26,28,32,37,-7,-49,-52,-55,-57,-58 14 16,22,23,29,30,36,-1,-4,-20,-21,-27,-34,-50,-54,-57,-63; 2,5,12,19,40,45,49,60,-8,-10,-32,-37,-38,-42,-44,-47 15 21,27,34,43,48,52,59,-2,-3,-20,-22,-28,-29,-35,-36,-41; 1,5,11,12,18,19,25,-9,-10,-37,-42,-47,-53,-56,-58,-62 16 13,24,27,31,34,42,47,-2,-5,-22,-23,-26,-28,-32,-60,-61; 3,6,8,10,14,55,57,-9,-11,-30,-40,-43,-45,-48,-53,-56 17 13,20,23,30,52,58,-3,-5,-21,-22,-24,-28,-46,-50,-51,-54; 1,2,6,9,33,38,39,44,-10,-11,-31,-37,-40,-45,-59,-62 18 15,20,22,26,47,51,-4,-6,-23,-24,-25,-27,-49,-50,-52,-61; 2,3,7,8,36,37,39,56,-11,-12,-35,-38,-40,-43,-57,-59 19 9,10,14,15,20,50,53,-5,-7,-25,-45,-48,-49,-51,-52,-55; 4,28,32,35,37,38,42,-12,-13,-30,-31,-34,-36,-39,-59,-60 20 59,-1,-2,-8,-26,-32,-33,-37,-38,-39,-42,-44,-47,-53,-56,-58; 5,11,12,17,18,19,24,25,31,40,45,49,-9,-14,-15,-63 21 13,23,29,30,35,36,41,58,-1,-5,-8,-26,-27,-32,-37,-59; 2,6,9,15,49,-11,-14,-17,-34,-43,-46,-48,-50,-51,-54,-63 22 13,27,34,52,-2,-3,-5,-26,-28,-29,-37,-40,-45,-53,-56,-62; 1,4,6,14,18,25,38,44,57,-15,-16,-17,-36,-46,-51,-60 23 31,34,47,51,54,-2,-6,-12,-29,-30,-33,-35,-38,-40,-43,-53; 3,7,8,10,14,17,21,37,-9,-16,-18,-49,-52,-58,-60,-61 24 20,26,30,32,-3,-6,-12,-27,-29,-31,-35,-40,-55,-57,-59,-62; 1,2,7,9,16,39,44,48,56,-10,-17,-18,-42,-46,-49,-50 25 15,20,22,26,29,33,53,-4,-7,-11,-30,-31,-34,-54,-57,-59; 2,3,8,37,42,46,-13,-18,-19,-41,-44,-45,-48,-49,-52,-61 19 20 DRAGOMIR Ž. –D OKOVIĆ 26 10,13,35,41,-1,-2,-5,-8,-32,-33,-43,-46,-48,-51,-60,-63; 4,6,18,24,25,31,49,57,-16,-20,-21,-22,-37,-39,-53,-56 27 30,36,41,45,-1,-6,-8,-12,-33,-34,-38,-42,-43,-46,-50,-63; 2,7,9,15,16,22,28,56,-14,-18,-21,-24,-52,-55,-57,-59 28 8,12,13,19,27,33,34,39,52,55,-4,-9,-10,-11,-32,-35; 1,6,-15,-16,-17,-22,-37,-40,-41,-45,-50,-53,-54,-56,-61,-62 29 11,32,34,54,-2,-3,-6,-12,-33,-35,-36,-42,-46,-49,-58,-60; 1,4,7,14,21,25,44,48,-15,-22,-23,-24,-40,-53,-55,-62 30 -2,-7,-16,-19,-36,-39,-41,-44,-45,-48,-49,-52,-56,-58,-60,-61; 3,8,10,14,17,21,24,27,37,42,46,50,-9,-13,-23,-25 31 20,26,32,33,38,43,53,-3,-7,-17,-19,-34,-36,-41,-45,-49; 1,2,9,16,23,-10,-13,-24,-25,-47,-51,-54,-55,-57,-59,-62 32 12,13,19,55,-1,-4,-9,-11,-14,-37,-38,-50,-53,-54,-56,-61; 3,6,24,29,31,36,49,-16,-20,-21,-26,-28,-43,-44,-48,-63 33 10,11,17,41,45,54,-1,-2,-6,-8,-12,-38,-39,-52,-53,-55; 4,7,25,28,31,-20,-23,-26,-27,-29,-42,-43,-46,-58,-60,-63 34 53,-1,-7,-8,-19,-21,-39,-44,-47,-48,-51,-52,-54,-55,-57,-59; 2,9,15,16,22,23,28,29,35,40,-13,-14,-25,-27,-31,-63 35 8,19,34,39,-4,-6,-9,-10,-18,-38,-40,-41,-50,-53,-61,-62; 1,5,7,21,26,48,51,-15,-23,-24,-28,-29,-42,-49,-57,-58 36 11,18,32,38,43,-2,-3,-7,-19,-22,-39,-41,-47,-51,-55,-62; 1,4,14,21,27,50,-13,-15,-29,-30,-31,-45,-49,-56,-58,-60 37 12,13,18,19,23,25,30,-3,-14,-15,-17,-26,-28,-42,-62,-63; 6,49,52,-8,-10,-20,-21,-22,-32,-46,-47,-50,-51,-53,-54,-56 38 17,19,22,45,-1,-4,-6,-9,-14,-18,-42,-43,-44,-57,-58,-63; 3,5,7,31,36,51,-20,-23,-27,-32,-33,-35,-50,-52,-53,-61 39 10,11,17,18,24,-1,-2,-7,-8,-19,-26,-44,-56,-58,-60,-63; 4,28,35,40,50,-13,-20,-30,-33,-34,-36,-47,-48,-51,-52,-55 EXPLICIT CAYLEY TRIPLES IN REAL FORMS OF E7 40 8,14,20,34,39,44,47,-5,-16,-17,-18,-28,-29,-43,-45,-49; 1,7,13,-11,-12,-22,-23,-24,-35,-46,-50,-53,-59,-60,-61,-62 41 8,43,46,-4,-7,-9,-10,-25,-28,-44,-45,-49,-54,-56,-57,-58; 1,5,12,21,26,27,33,-13,-15,-30,-31,-35,-36,-47,-61,-62 42 19,25,30,45,48,-3,-6,-14,-15,-24,-33,-35,-46,-47,-50,-53; 5,7,11,16,-8,-10,-20,-27,-29,-37,-38,-55,-57,-58,-62,-63 43 47,-1,-5,-16,-18,-21,-32,-33,-46,-48,-50,-52,-53,-59,-60,-61; 3,7,10,13,15,31,36,41,-11,-12,-23,-26,-27,-38,-40,-63 44 17,22,24,29,46,-1,-4,-7,-9,-14,-25,-32,-47,-48,-52,-61; 3,5,12,40,-13,-20,-30,-34,-38,-39,-41,-54,-56,-57,-58,-63 45 8,14,20,-5,-7,-16,-17,-25,-28,-36,-48,-49,-59,-60,-61,-62; 1,6,27,33,38,42,-11,-19,-22,-30,-31,-40,-41,-51,-54,-56 46 9,25,30,-3,-5,-21,-22,-24,-33,-37,-40,-50,-51,-55,-62,-63; 4,7,13,41,44,-8,-12,-17,-26,-27,-29,-42,-43,-53,-59,-60 47 -3,-7,-14,-15,-31,-37,-39,-41,-51,-54,-55,-56,-57,-58,-62,-63; 5,11,12,16,18,23,40,43,-8,-10,-13,-20,-34,-36,-42,-44 48 24,29,35,-1,-5,-7,-16,-21,-25,-32,-39,-51,-52,-54,-56,-63; 3,6,10,15,42,-11,-19,-26,-30,-34,-43,-44,-45,-59,-60,-61 49 8,14,20,21,26,32,37,-6,-12,-13,-23,-24,-25,-35,-36,-52; 1,-18,-19,-29,-30,-31,-40,-41,-45,-55,-57,-58,-59,-60,-61,-62 50 30,36,39,-4,-10,-11,-21,-24,-28,-37,-38,-40,-53,-54,-57,-59; 2,7,13,19,-14,-17,-18,-27,-32,-35,-42,-43,-46,-61,-62,-63 51 9,35,38,-3,-5,-7,-21,-22,-31,-37,-39,-45,-54,-55,-59,-60; 4,6,18,23,-8,-17,-19,-26,-34,-36,-46,-47,-48,-56,-62,-63 52 37,-1,-6,-12,-13,-23,-25,-27,-38,-39,-55,-57,-58,-59,-60,-61; 3,10,15,17,22,28,-18,-19,-30,-33,-34,-43,-44,-48,-49,-63 53 -2,-15,-16,-26,-28,-29,-37,-38,-40,-46,-56,-58,-60,-61,-62,-63; 7,13,19,25,31,34,-9,-20,-22,-23,-32,-33,-35,-42,-43,-50 21 22 DRAGOMIR Ž. –D OKOVIĆ 54 12,-4,-7,-10,-11,-21,-28,-31,-37,-44,-45,-56,-57,-61,-62,-63; 2,6,23,29,33,-14,-17,-25,-32,-34,-41,-47,-48,-50,-51,-59 55 9,11,16,-3,-6,-12,-13,-27,-29,-31,-39,-42,-49,-57,-62,-63; 4,28,32,-8,-19,-24,-33,-34,-36,-46,-47,-51,-52,-58,-59,-60 56 12,18,24,27,-2,-7,-15,-16,-26,-28,-36,-37,-44,-45,-51,-58; 6,-9,-20,-22,-30,-32,-39,-41,-47,-48,-53,-54,-60,-61,-62,-63 57 16,22,26,-4,-6,-10,-13,-18,-27,-31,-35,-42,-44,-49,-58,-59; 2,5,-14,-24,-25,-34,-38,-41,-47,-50,-52,-54,-55,-61,-62,-63 58 -2,-6,-13,-15,-23,-33,-35,-36,-42,-44,-49,-55,-60,-61,-62,-63; 5,11,17,21,-9,-20,-29,-30,-38,-39,-41,-47,-52,-53,-56,-57 59 -5,-17,-18,-19,-27,-31,-40,-46,-48,-49,-54,-55,-60,-61,-62,-63; 2,9,15,20,-11,-12,-21,-24,-25,-34,-43,-45,-50,-51,-52,-57 60 10,14,-2,-5,-19,-22,-23,-33,-36,-40,-46,-48,-49,-55,-56,-61; 4,-12,-16,-26,-29,-30,-39,-43,-45,-51,-52,-53,-58,-59,-62,-63 61 -4,-9,-11,-23,-25,-28,-38,-40,-41,-48,-49,-50,-56,-57,-62,-63; 3,8,-16,-18,-30,-32,-35,-43,-44,-45,-52,-53,-54,-58,-59,-60 62 -3,-15,-17,-28,-29,-31,-40,-41,-42,-49,-50,-51,-56,-57,-60,-63; 1,-10,-22,-24,-35,-36,-37,-45,-46,-47,-53,-54,-55,-58,-59,-61 63 -1,-8,-20,-21,-32,-33,-34,-42,-43,-44,-50,-51,-52,-56,-57,-60; -14,-26,-27,-37,-38,-39,-46,-47,-48,-53,-54,-55,-58,-59,-61,-62 EXPLICIT CAYLEY TRIPLES IN REAL FORMS OF E7 23 References [1] D. Barbasch and M.R. Sepanski, Closure ordering and the Kostant-Sekiguchi correspondence, Proc. Amer. Math. Soc., 126 (1998), 311-317. [2] D.H. Collingwood and W.M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold, New York, 1992. [3] D. Ž. –D oković , On automorphisms and adjoint orbits of real semisimple Lie algebras, J. Algebra, 94 (1985), 19-29. [4] , Classification of nilpotent elements in simple exceptional real Lie algebras of inner type and description of their centralizers, J. Algebra, 112 (1988), 503-524. [5] , Classification of nilpotent elements in simple real Lie algebras E6(6) and E6(−26) and description of their centralizers, J. Algebra, 116 (1988), 196-207. [6] , Note on rational points in nilpotent orbits of semisimple groups, Indag. Math., 9 (1998), 31-34. [7] , Explicit Cayley triples in real forms of G2 , F4 , and E6 , Pacific J. Math., 184 (1998), 231-255. [8] D.R. King, The component groups of nilpotents in exceptional simple real Lie algebras, Comm. Algebra, 20 (1992), 219-284. [9] J.F. Kurtzke, Centers of centralizers in reductive algebraic groups, Comm. Algebra, 19 (1991), 3393-3410. [10] R. Steinberg, Lectures on Chevalley groups, Yale University, 1967. Received January 14, 1998 and revised April 24, 1998. The author was supported in part by the NSERC Grant A-5286. University of Waterloo Waterloo, Ontario N2L 3G1 Canada E-mail address: [email protected] PACIFIC JOURNAL OF MATHEMATICS Vol. 191, No. 1, 1999 ON THE INDEX FORMULA FOR SINGULAR SURFACES B. Fedosov, B.-W. Schulze, and N. Tarkhanov The index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points contains the Atiyah-Singer integral as well as two additional terms. One of the two is the ‘eta’ invariant defined by the conormal symbol, and the other term is explicitly expressed via the principal and subprincipal symbols of the operator at conical points. The aim of this paper is an explicit description of the contribution of a conical point for higher-order differential operators. We show that changing the origin in the complex plane reduces the entire contribution of the conical point to the shifted ‘eta’ invariant. In turn this latter is expressed in terms of the monodromy matrix for an ordinary differential equation defined by the conormal symbol. Introduction. In [FST97] we proved the following index formula for elliptic pseudodifferential operators on a two-dimensional manifold with a conical point: Z Z τ =1 i 1 (0.1) ind A = tr σ0−1 σsub τ =−1 dξdx, AS (A) − η(Ac ) + 2 2 4π S1 ×R S∗M M being the manifold in question whose cross-section close to the conical point is identified with the unit circle S1 . The index is evaluated for A acting on weighted Sobolev spaces on M as H s,γ (M, E 0 ) → H s−m,γ (M, E 1 ), where E 0 and E 1 are C ∞ vector bundles over the smooth part of M which behave properly when approaching the conical point. The first term on the right-hand side of this formula is the Atiyah-Singer integral derived from the principal interior symbol σ0 of A and the curvature forms Ω0 and Ω1 of the bundles E 0 and E 1 , respectively. We have 1 1 1 −1 −1 3 0 −1 1 AS (A) = 2 tr (σ0 ∂σ0 ) − tr Ω σ0 ∂σ0 + Ω ∂σ0 σ0 . 4π 6 2 The weight exponent γ enters only the second term on the right side of (0.1) which is known as the ‘eta’ invariant of the conormal symbol Ac of A 25 26 B. FEDOSOV, B.-W. SCHULZE, AND N. TARKHANOV at the conical point. More precisely, d 1 −1 0 −1 0 η(Ac ) = − Tr Ac (τ + iγ)Ac (τ + iγ) − iγ (A (τ + iγ)Ac (τ + iγ)) , πi dτ c Tr being a regularised trace (cf. Melrose [Mel95]). Both these terms occur in the Atiyah-Patodi-Singer formula for the index of Dirac operators (cf. [APS75]). In contrast to this latter formula, (0.1) contains the additional third term which does not vanish even for the Cauchy-Riemann operator on the plane. This summand also depends on the conormal symbol Ac (τ ) only because the principal symbol σ0 and the so-called subprincipal symbol σsub = σ1 + i ∂ 2 σ0 2 ∂x∂ξ are evaluated at the conical point. Here σ1 means the homogeneous component of degree deg σ0 − 1 of the complete symbol of Ac (τ ). Of course, formula (0.1) is still true for manifolds with several conical points. A slight change we have to do is that the ‘eta’ invariant and the additional terms should be summed up over all conical points of M . The aim of this paper is an explicit description of the contribution of a conical point for elliptic differential operators. To this end we show first that by changing the origin in the complex τ -plane we can make the third term to vanish reducing the whole contribution of the conical point to the shifted ‘eta’ invariant. The new origin τ0 which we refer to as the centre is the root of the linear equation τ =1 Z −1 ∂σ0 (0.2) τ0 + σsub dξdx = 0. tr σ0 ∂τ S1 ×R τ =−1 The next goal is to express the ‘eta’ invariant in terms of the monodromy matrix M (τ ) for an ordinary differential equation defined by the conormal symbol Ac (τ ). We introduce a phase function 1 log det M (τ ) + M −1 (τ ) − 2 2 which is an analytic function of τ with logarithmic ramification points. Then our final index theorem reads Z 1 (0.3) ind A = AS(A) + ∆Γ,τ0 ϕ(τ ) 2πi S∗ M ϕ(τ ) = where ∆Γ,τ0 ϕ(τ ) denotes the variation of the phase function along a suitable contour defined by the weight line Γ and the centre τ0 (Theorem 3.1). In some particular cases we may say more about the variation ∆Γ,τ0 ϕ(τ ). For example, if the function f (τ ) = det(M (τ ) + M −1 (τ ) − 2) is even with INDEX FORMULA FOR SINGULAR SURFACES 27 respect to τ0 , that is f (τ0 − T ) = f (τ0 + T ), then the second term in (0.3) may be calculated in terms of zeros of f (τ ) and turns out to be half-integer (Theorem 3.2). Thinking over these properties we have come to a generalisation of the symmetry conditions used in [SSS97]. A detailed treatment of this symmetry in the higher-dimensional case will be given in a forthcoming paper. 1 Finally, we show that the above integrality of πi ∆Γ,τ0 ϕ(τ ) holds for any first-order elliptic system, no matter whether the symmetry condition is fulfilled or not. To this end we investigate the asymptotical behaviour of solutions and the monodromy matrix when <τ → ±∞ and =τ remains bounded. Although there exists vast literature on this topic, we have not found the desired facts and were forced to prove them. The proof uses the ideas of Faddeev and Takhtajan [FT87] for the non-linear Schrödinger equation. Acknowledgments. The authors wish to express their gratitude to M. Lesch for drawing the authors’ attention to the paper [BFK91] where the monodromy matrix appeared in an expression for the determinant of an elliptic differential operator on a circle. 1. The existence of the centre. Recall that the neighbourhood of a conical point is treated as a cylindrical end with coordinates t ∈ R+ and x ∈ R mod (2π). Since any complex vector bundle over a circle is trivial, we may assume that E 0 ∼ = E1 ∼ = Cr over the cylindrical end and, for given trivialisations, the connection one-forms Γ0 , Γ1 are equal to 0. The conormal symbol of an m th order differential operator has the form (1.1) Ac (τ ) = am (x) ∂m ∂ m−1 + a (x, τ ) + . . . + a0 (x, τ ). m−1 ∂xm ∂xm−1 So, it is an ordinary differential operator on a circle whose coefficients ak (x, τ ) = m−k X ak,l (x)τ l l=0 are polynomials in τ of degree m − k. Thus, the principal symbol of the operator A restricted to the boundary is σ0 (A) = m X k=0 am−k,k (x) τ k (iξ)m−k 28 B. FEDOSOV, B.-W. SCHULZE, AND N. TARKHANOV and for the lower-order term we have m−1 X σ1 (A) = am−1−k,k (x) τ k (iξ)m−1−k . k=0 The interior ellipticity means that σ0 (A) is an invertible matrix for any real (ξ, τ ) 6= (0, 0); in particular, the coefficient am (x) in (1.1) is an invertible matrix-valued function on a circle. Without loss of generality we assume that am (x) ≡ 1, otherwise we change the frame in E 1 using am as a transition matrix. Replacing τ by τ + τ0 in (1.1), we see that the shift by τ0 in the complex τ -plane does not change the principal symbol σ0 , while for σ1 we have a new expression ∂σ0 σ f1 = σ1 + τ0 . ∂τ The subprincipal symbol σsub obeys the same rule ∂σ0 σg τ0 . sub = σsub + ∂τ Thus, after shifting we obtain a new additional term in (0.1) proportional to the left-hand side of (0.2). The following theorem guaranties a unique solvability of the linear equation (0.2). Theorem 1.1. For any elliptic differential operator A, τ =1 Z −1 ∂σ0 (1.2) tr σ0 dxdξ 6= 0. ∂τ τ =−1 S1 ×R Proof. Denoting det σ0 by f (x, τ, ξ), we have ∂σ0 ∂f tr σ0−1 = f −1 . ∂τ ∂τ From ellipticity we deduce that the roots of the polynomial f = f (τ, ξ) for fixed real τ form two disjoint sets corresponding to the upper and lower halfplanes. The integrand in (1.2) is a rational function in ξ decaying as O(|ξ|−2 ) when ξ → ∞. Thus, integrating over ξ, we may replace the real axis by a closed contour c± consisting of a large semicircle in the upper (lower) halfplane and its diameter and surrounding all the poles in the corresponding half-plane. By the Euler theorem for homogeneous functions, 1 −1 ∂f −1 ∂f f = mr − ξf , ∂τ τ ∂ξ so that τ =1 −1 ∂f −1 ∂f −1 ∂f f = 2mr − ξf − ξf ∂τ ∂ξ ∂ξ τ =−1 τ =1 τ =−1 INDEX FORMULA FOR SINGULAR SURFACES 29 and the residue theorem yields τ =1 Z X X + −1 ∂f f dξ = −2πi ξ + ξk− k ∂τ τ =−1 R where ξk± are the roots of the equation f (±1, ξ) = 0 in the upper half-plane. Since f (±1, ξ) = 0 is equivalent to f (1, ±ξ) = 0, we see that at least one set ξ + or ξ − is not empty. Thus, X X = ξk+ + ξk− > 0, proving the theorem. Remark 1.2. Our proof uses essentially the fact that f (τ, ξ) is a homogeneous polynomial. Clearly, for rational homogeneous functions f (τ, ξ) having no zeros and poles on the real axis =ξ = 0 the theorem is not true. 2. The Green function and the monodromy matrix. The operator d dτ A−1 c d Ac (τ ) dτ is a pseudodifferential operator of order −2 on the circle, thus it belongs to the trace class. Its trace may be explicitly calculated in terms of the so-called monodromy matrix. Consider the ordinary differential equation (2.1) Ac (τ )u = u(m) (x) + am−1 (x, τ )u(m−1) (x) + . . . + a0 (x, τ )u(x) = 0. Its solutions form a linear space of dimension mr. Since the coefficients are 2π-periodic functions, the shift u(x) 7→ u(x + 2π) defines a linear transformation M of the space of solutions called the monodromy. Theorem 2.1. The monodromy transformations M (τ ), M −1 (τ ) are entire functions in τ , and 1 d2 d −1 d Tr Ac Ac (τ ) = log det M (τ ) + M −1 (τ ) − 2 . 2 dτ dτ 2 dτ Proof. Any solution u(x) is uniquely defined by the vector of its Cauchy data u(x) u0 (x) (2.2) ~u(x) = ... (m−1) u (x) at some point x0 . The monodromy carries the vector ~u(x0 ) to ~u(x0 +2π) and we may calculate the monodromy matrix as follows. Consider the Wronsky 30 B. FEDOSOV, B.-W. SCHULZE, AND N. TARKHANOV matrix U (x, x0 , τ ) consisting of linearly independent vector-valued functions (2.2) normalised by the initial condition (2.3) U |x=x0 = U (x0 , x0 , τ ) = 1. Then, M (τ ) = U (x0 + 2π, x0 , τ ), M −1 (τ ) = U (x0 − 2π, x0 , τ ). The Wronsky matrix satisfies a first-order differential equation A(τ )U = 0 (2.4) where A(τ ) is given by the block matrix d/dx −1 0 d/dx ... A(τ ) = (2.5) ... 0 0 a0 (x, τ ) a1 (x, τ ) ... ... ... ... ... 0 0 ... −1 am−1 (x, τ ) + d/dx . Since the coefficients are polynomials in τ , the solution U (x, x0 , τ ) is a holomorphic function in τ ∈ C and so are M (τ ) and M −1 (τ ). We will express the operator A0c (τ )A−1 c (τ P) through the first order operator (2.5). Introduce the notion B for the sum of diagonal blocks of a block matrix B. Lemma 2.2. The following equality holds X A0c (τ )A−1 A0 (τ )A−1 (τ ). c (τ ) = Proof. To find A−1 (τ ), write the equation A~u(τ ) = ~v for ~u in components du0 − u1 = v0 , dx du1 − u2 = v1 , dx ... ... ... dum−1 + am−1 um−1 + . . . + a0 u0 dx = vm−1 . Eliminating u1 = u2 = du0 − v0 , dx d du0 − v0 − v1 dx dx and so on, we obtain an equation for u0 of the form (d/dx)m u0 + am−1 (d/dx)m−1 u0 + . . . + a0 u0 = w INDEX FORMULA FOR SINGULAR SURFACES 31 where w is a known function, namely a linear combination of v0 , v1 , . . . , vm−1 and their derivatives. Thus, u0 = A−1 c (τ )w, and moving backward we find successively u1 , u2 , . . . , um−1 . The most simple expression we have in the case when v0 = v1 = . . . = vm−2 = 0. Then u0 = A−1 c (τ )vm−1 , u1 = (d/dx) A−1 c (τ )vm−1 , ... ... ... um−1 = (d/dx)m−1 A−1 c (τ )vm−1 . It follows that A−1 exists exactly when A−1 c does and ∗ ... ∗ A−1 c (τ ) ∗ . . . ∗ (d/dx) A−1 c (τ ) A−1 (τ ) = ... ... ... ... ∗ . . . ∗ (d/dx)m−1 A−1 c (τ ) where ∗ means any expression whose explicit form is irrelevent. Next, 0 0 ... 0 ... ... ... ... , A0 (τ ) = (2.6) 0 0 ... 0 a00 a01 . . . a0m−1 the prime meaning the derivation in τ , so that 0 0 ... 0 . . . . . . . . . . .. A0 (τ )A−1 (τ ) = 0 0 ... 0 ∗ ∗ . . . A0c (τ )A−1 c (τ ) proving the lemma. Now we find A−1 (τ ) in another way. Let x vary in the closed interval [0, 2π] and let U (x, τ ) be the Wronsky matrix satisfying (2.3) at x0 = 0. The operator A−1 (τ ), when considered on periodic functions on [0, 2π], is an integral operator whose kernel G(x, y, τ ) (the Green function) is a periodic solution of the equation A(τ )G(x, y, τ ) = δ(x − y). We treat x as an argument while y ∈ [0, 2π] is considered as a parameter, δ being the Dirac δ-function. This equation means that G satisfies the homogeneous equation on [0, y) and (y, 2π], whence G(x, y, τ ) = U (x, τ )C− , G(x, y, τ ) = U (x, τ )C+ , for x ∈ [0, y), for x ∈ (y, 2π], 32 B. FEDOSOV, B.-W. SCHULZE, AND N. TARKHANOV the matrices C± being independent of x. To produce the δ-function, these matrices should satisfy the relation C+ − C− = U −1 (y, τ ) while periodicity yields C− = U (2π, τ )C+ = M (τ )C+ . Solving this system, we get a usual expression for the Green function, namely U (x, τ )(1 − M (τ ))−1 U −1 (y, τ ), x ∈ [0, y), G(x, y, τ ) = −1 −1 U (x, τ )M (τ )(1 − M (τ )) U (y, τ ), x ∈ (y, 2π], or equivalently G(x, y, τ ) = 1 U (x, τ )(1 + M (τ ))(1 − M (τ ))−1 U −1 (y, τ ) 2 1 + sgn (x − y)U (x, τ )U −1 (y, τ ). 2 0 Using Lemma 2.2, we conclude that the operator (A0c (τ )A−1 c (τ )) has the kernel 1 X d (2.7) A0 (τ )U (x, τ )(1 + M (τ ))(1 − M (τ ))−1 U −1 (y, τ ) 2 dτ X d 1 + sgn (x − y) A0 (τ )U (x, τ )U −1 (y, τ ) . 2 dτ The second term vanishes at x = y because from (2.6) X d A0 (τ ) = a00m−1 (τ ) dτ = 0 0 since am−1 is a linear function in τ . To calculate the trace of A0c (τ )A−1 c (τ ) (which belongs to the trace class), we put x = y in (2.7), take the matrix trace and integrate over [0, 2π]. The second term in (2.7) may be dropped and we obtain 0 Tr A0c (τ )A−1 c (τ ) Z 2π ∂G = (x, x, τ ) dx tr ∂τ 0 Z 2π 1 d −1 = tr (1 + M (τ ))(1 − M (τ )) U −1 (x, τ )A0 (x, τ )U (x, τ ) dx. 2 dτ 0 To complete the proof of Theorem 2.1, we need the following lemma. INDEX FORMULA FOR SINGULAR SURFACES 33 Lemma 2.3. We have M −1 (τ )M 0 (τ ) = − 2π Z U −1 (x, τ )A0 (x, τ )U (x, τ ) dτ. 0 Proof. Differentiating (2.4) in τ , we obtain A(τ )U 0 + A0 (τ )U = 0 (2.8) with an initial condition U 0 (x, τ ) |x=0 = 0, where the prime means derivation in τ . To find U 0 , we apply a variation of constants to (2.8) looking for U 0 in the form U V . Then (2.8) yields U dV + A0 (τ )U = 0, dx Z x so that V (x) = − U −1 (y, τ )A0 (y, τ )U (y, τ ) dy 0 and U 0 (x, τ ) = U (x, τ ) Z x U −1 (y, τ )A0 (y, τ )U (y, τ ) dy. 0 Taking x = 2π yields the desired identity. Now 0 1 d Tr A0c (τ )A−1 = − tr (1 + M )(1 − M )−1 M −1 M 0 c (τ ) 2 dτ d 1 d = tr (M − 1)−1 M 0 − tr M −1 M 0 dτ 2 dτ 1 d2 = log det(M − 1)2 M −1 2 dτ 2 which is precisely (2.1). 3. The index formula. Combining the results of Sections 1 and 2, we obtain a simple interpretation of the boundary terms in the index formula (0.1). We also introduce a symmetry condition generalising that of [SSS97]. It allows one to simplify further the boundary term reducing it to the number of poles of A−1 c in a strip. Consider two horizontal lines Γ, Γ0 in the complex τ -plane, Γ0 passing through the centre τ0 . In the strip between these lines the operator A−1 c (τ ) has a finite number of poles. In particular, for |<τ | > T0 1 there are no poles at all. Consider a contour starting at the point τ0 − T with T > 0 34 B. FEDOSOV, B.-W. SCHULZE, AND N. TARKHANOV large enough, so that |<(τ0 − T )| > T0 , then going along Γ in the region where |<τ | < T0 , and terminating at the point τ0 + T (see Fig. 1). 6 Γ τ0 −T τ0 u u −T0 τ0 +T u -Γ0 T0 Figure 1. Variation of ϕ(t) along Γ. The function 1 log det(M + M −1 − 2) 2 is analytic with ramification points at zeros of ϕ(τ ) = f (τ ) = eϕ(τ ) = det1/2 (M + M −1 − 2) det(M − 1) = . det1/2 M Clearly, the zeros of det(M (τ ) − 1) are the poles of A−1 c (τ ). Denote by ϕ(τ0 + T ) − ϕ(τ0 − T ) the variation of ϕ(τ ) along the contour described above and set (3.1) ∆Γ,τ0 ϕ(τ ) = lim (ϕ(τ0 + T ) − ϕ(τ0 − T )) . T →∞ With this notation we have the following theorem. Theorem 3.1. Let τ0 be the centre and M (τ ) be the monodromy matrix of the ordinary differential operator Ac (τ ) on the circle. Then Z 1 (3.2) ind A = AS(A) + ∆Γ,τ0 ϕ(τ ). 2πi ∗ S M Proof. Let us consider the isomorphisms of the bundles E 0 , E 1 consisting in multiplication by exp(iτ0 t). The local expressions for the operator A = Ac (−i∂/∂t) in cylindrical charts change to ∂ −iτ0 t iτ0 t = Ac −i + τ0 . e Ae ∂t So, its conormal symbol changes to Ac (τ + τ0 ). INDEX FORMULA FOR SINGULAR SURFACES 35 If τ0 is the centre, then according to Section 1 the boundary term consists of 1 1 0 Tr A−1 − η(Ac (τ + τ0 )) = c (τ + τ0 )Ac (τ + τ0 ). 2 2πi It is sufficient to compute (3.3) for τ0 = 0. Consider ∂ −1 ∂ 0 0 A−1 (τ )A (τ ) − iγ A (τ )A (τ ) . Q(τ ) = Tr c c c ∂τ ∂τ c (3.3) By Theorem 2.1, this quantity is equal to ∂2 ∂ ϕ(τ ) − iγ ϕ(τ ) . ∂τ 2 ∂τ According to the definition of Tr (see Melrose [Mel95]) and η(Ac ), we obtain Z T Z τ1 1 − η(Ac ) = lim dτ1 Q(τ + iγ)dτ, T →∞ −T 2 0 the right-hand side being understood as a constant term in the asymptotic expansion when T → ∞. Thus, τ =T +iγ 1 ∂ 1 lim ϕ(τ ) − iγ ϕ(τ ) − η(Ac ) = 2 2πi T →∞ ∂τ τ =−T +iγ and the variation of ϕ(τ ) is taken along the weight line Γ (for (∂/∂τ )ϕ(τ ), the variation does not depend on the path). In the region |<τ | > T0 where ϕ(τ ) is holomorphic in the strip between Γ and the real axis, we may use the Taylor formula, thus obtaining ∂ ϕ(τ ) = ϕ(τ − iγ) + R2 (τ, γ) ∂τ where R2 (τ, γ) is a remainder term which tends to 0 for <τ → ±∞ and |=τ | ≤ C. Hence it follows that ϕ(τ ) − iγ 1 1 η(Ac ) = lim (ϕ(T ) − ϕ(−T )) , 2 2πi T →∞ the variation is taken along the contour in Fig. 1 with τ0 = 0. This completes the proof. − There are important particular cases when the variation (3.1) can be calculated by the residue theorem. Theorem 3.2. Let f 2 (τ ) = det(M (τ ) + M −1 (τ ) − 2) be an even function with respect to the centre τ0 , that is (3.4) f 2 (τ0 + T ) = f 2 (τ0 − T ) 36 B. FEDOSOV, B.-W. SCHULZE, AND N. TARKHANOV for any T . Then Z (3.5) 1 AS(A) + p + q ind A = 2 ∗ S M sgn (=τ0 − γ) where p is the number of zeros of f (τ ) (counted along with their multiplicities) in the strip between Γ and Γ0 , and q is the number of zeros on the line Γ0 . Proof. To be specific, let =τ0 < γ. Consider a closed contour l = l1 ∪ l2 where l1 is the contour on Fig. 1 and l2 goes along the line Γ0 bypassing the zeros lying on Γ0 along small semicircles (see Fig. 2). 6 -l1 Γ l2 u u 2τ0 −τ1 τ0 u τ1 -Γ0 Figure 2. The contour l = l1 ∪ l2 . Clearly, 1 ∆Γ,τ0 ϕ(τ ) = 2πi 1 ∆l1 ϕ(τ ) 2πi 1 ∆l2 ϕ(τ ). = −p − 2πi We next observe that the variation of ϕ(τ ) along l2 is equal to the sum of variations along all the semicircles. Indeed, the variations along the segments of Γ0 cancel because of (3.4). When the radii of the semicircles tend to 0, the variations along them tend to πi times the number q of zeros on Γ0 counted together with their multiplicities. This is the desired conclusion. Since the result is very simple, it is desirable to have simple sufficient conditions for (3.4) to be fulfilled. One of these is the symmetry condition of [SSS97] for the conormal symbol: there exist isomorphisms v0 (x) and v1 (x) of the bundles E 0 and E 1 , such that (3.6) Ac (τ0 − T ) = v1 (x) Ac (τ0 + T ) v0 (x) for each real T . Roughly speaking (3.6) means that the symmetry transformation τ 7→ 2τ0 − τ acts on Ac (τ ) by an automorphism of the algebra of differential operators on S1 induced by isomorphisms of the bundles E 0 , E 1 . INDEX FORMULA FOR SINGULAR SURFACES 37 We introduce more general symmetry conditions including automorphisms generated by changes of variables. Definition 3.3. The conormal symbol Ac (τ ) is called symmetric (with respect to the centre τ0 ) if there exist a diffeomorphism g : S1 → S1 and bundle isomorphisms v0 : g ∗ E 0 → E 0 , v1 : g ∗ E 1 → E 1 such that (3.7) Ac (τ0 − T ) = (g −1 )∗ v1−1 Ac (τ0 + T )v0 g ∗ . The definition gains in interest if we realise that differential operators with symmetric conormal symbols meet the condition of Theorem 3.2. Proposition 3.4. For symmetric conormal symbols (3.4) holds. Proof. A diffeomorphism g : S1 → S1 is defined by a monotone function g(x), x ∈ R1 , such that g(x + 2π) = g(x) ± 2π where the sign ‘+’ means that g preserves the orientation while ‘−’ corresponds to diffeomorphisms reversing the orientation. If U (x, τ0 − T ) is the Wronsky matrix for Ac (τ0 − T ), then by (3.7) we have U (x, τ0 + T ) = v0 (x) U (g(x), τ0 − T ) v0−1 (x). Taking x = 2π, we obtain M (τ0 + T ) = v0 (0) M ±1 (τ0 − T ) v0−1 (0). Thus, in the case of orientation-preserving diffeomorphisms g (in particular, under the symmetry condition (3.6)) we have M (τ0 + T ) = v0 (0) M (τ0 − T ) v0−1 (0), while an orientation-reversing diffeomorphism g yields M (τ0 + T ) = v0 (0) M −1 (τ0 − T ) v0−1 (0). Both these properties imply (3.4) and thus (3.5). Consider some examples illustrating Proposition 3.4. Example 3.5. Let τ0 = 0; g : x 7→ −x and v0 = 1, v1 = (−1)m . Then (3.7) written for X ∂l Ac (τ ) = ak,l (x) τ k l ∂x k+l≤m just amounts to the fact that ak,l (−x) = (−1)m+k+l ak,l (x). 38 B. FEDOSOV, B.-W. SCHULZE, AND N. TARKHANOV In other words, the coefficients are even matrix functions if k + l and m have the same parity, and odd functions otherwise. In particular, constant coefficients will do, provided that ak,l = 0 for k + l 6≡ m (mod 2). Example 3.6. For a first-order scalar differential operator Ac (τ ) = d − a(x)τ − b(x) dx Proposition 3.4 always holds. Indeed, the monodromy is given by a scalar factor Z 2π M (τ ) = exp (a(x)τ + b(x)) dx 0 and the centre τ0 is the root of Equation (0.2) which in our case reduces to Z 2π (a(x)τ0 + b(x)) dx = 0. 0 Clearly, M (τ0 − T ) = M −1 (τ0 + T ). A particular case of this example is the Cauchy-Riemann operator on a Riemann surface with conical points. Remark 3.7. It is interesting that the index formula in the form (3.5) under symmetry condition (3.7) is valid in the general setting of pseudodifferential operators on a higher-dimensional manifold with conical singularities. The proof using the ideas of [SSS97] and the machinery of [FST97] will be given in a forthcoming paper. 4. First-order operators. Consider in more detail the case of a first-order matrix-valued operator (4.1) Ac (τ ) = d − A(x)τ − B(x). dx We will show that, similarly to Example 3.6, the centre is completely determined by the monodromy matrix, or rather by its asymptotic behaviour when <τ → ±∞ while =τ remains bounded. The asymptotics implies that the boundary contribution in the index formula (3.2) is half-integer provided the frames in E 0 , E 1 are chosen in an appropriate way. Consequently, the Atiyah-Singer term also has a half-integer value. The interpretation in terms of zeros as in (3.5) fails in general. We begin with a choice of frames in E 0 and E 1 . By the interior ellipticity, the spectrum of A(x) at any x ∈ S1 does not intersect the imaginary axis, INDEX FORMULA FOR SINGULAR SURFACES 39 so it consists of two disjoint parts in the right and left half-planes. The corresponding spectral projectors are given by the Cauchy integrals Z 1 (4.2) (ξ + iA(x))−1 dξ P± (x) = 2πi c± where the contours c± surround the spectrum in the corresponding halfplanes. These projectors depend smoothly on x defining a splitting of the trivial bundle Cr ∼ = E0 ∼ = E 1 into a direct sum of two subbundles. Like any complex bundle over a circle, these subbundles are trivial. It follows that we may choose a frame in Cr with a transition matrix C(x), so that 1 0 P+ (x) = C(x) C −1 (x), 0 0 0 0 P− (x) = C(x) C −1 (x) 0 1 and A−1 1 (x)A2 (x) = iC(x) a+ (x) 0 0 a− (x) C −1 (x), where a± (x) are (r± ×r± )-matrices having the spectra in the right (left) halfplane. Passing to new frames in E 0 , E 1 with the same transition matrix C(x), we reduce the matrix A(x) to a block-diagonal form a+ (x) 0 (4.3) A(x) = . 0 a− (x) Here the matrices a± have their spectra in the right (left) half-planes. The matrix B(x) changes to C −1 BC + C −1 dC dx and may be written in a block form b11 (x) b12 (x) (4.4) B(x) = b21 (x) b22 (x) corresponding to (4.3). Thus, we have reduced the conormal symbol Ac (τ ) to the canonical form (4.1), (4.3), (4.4) with a block-diagonal matrix A(x). Lemma 4.1. The centre τ0 is the root of the equation Z 2π (4.5) (tr (a+ (x)τ + b11 (x)) − tr (a− (x)τ + b22 (x))) dx = 0. 0 Proof. We have σ0 = iξ − A(x)τ ; σsub = −B(x), 40 B. FEDOSOV, B.-W. SCHULZE, AND N. TARKHANOV so Equation (0.2) reduces to Z 2π Z ∞ dx (ξ + iA(x))−1 − (ξ − iA(x))−1 (A(x)τ0 + B(x)) dξ = 0. tr 0 −∞ Integrating over ξ and using (4.2), we obtain Z 2π 1 0 tr (A(x)τ0 + B(x)) dx = 0, 0 1 0 which is precisely (4.5). 5. Lyapunov estimates. In this section we consider the so-called stable case when the part a+ (x) in (4.3) is absent. So, all the eigenvalues of A(x) = a− (x) have negative real parts. Such matrices will be called stable. The Wronsky matrix as a function of x is a solution of the Cauchy problem dU = (A(x)τ + B(x)) U, dx U |x=y = 1. The following theorem gives an estimate for the fundamental solution U (x, y, τ ) = U (x, τ )U −1 (y, τ ). Theorem 5.1. Let A(x) be a stable matrix. Then there exist constants C, d > 0 such that, for τ 1, (5.1) kU (x, y, τ )k ≤ C exp(−d (x − y) τ ), provided x ≥ y, where k · k means any matrix norm. Remark 5.2. In the case of constant coefficients A and B, estimate (5.1) is obvious, because the solutions can be expressed in terms of exponential functions. For variable coefficients it is not, however, so obvious (recall stable and unstable zones for the Schrödinger equation). The following necessary and sufficient condition of stability is due to Lyapunov (see e.g. [Gan86]). Lemma 5.3. A complex matrix A is stable if and only if there exists a Hermitian positive definite matrix X such that (5.2) A∗ X + XA = −1. Proof. If A is stable, so is A∗ . Hence both exp At and exp A∗ t are exponentially decaying as t → +∞. The matrix X may be defined by an explicit expression, namely Z ∞ (5.3) X= exp(A∗ t) exp(At)dt. 0 INDEX FORMULA FOR SINGULAR SURFACES 41 Indeed, Z ∗ ∞ A X + XA = 0 ∂ (exp(A∗ t) exp(At)) dt ∂t = −1. Conversely, from (5.2) it follows, for an eigenvector e of A with an eigenvalue λ, that (e, e) = −(XAe, e) − (Xe, Ae) = −(λ + λ̄)(Xe, e). Hence <λ < 0, as desired. Proof of Theorem 5.1. If A(x) is a smooth periodic function in x, then (5.3) shows that X(x) is also a smooth periodic function. In particular, there are bounds independent of x, for 0 < C1 ≤ X(x) ≤ C2 (5.4) in the sense of quadratic forms. Denoting the usual norm in Cr by kek = p (e, e), we define a new norm p kekX = (Xe, e) which is equivalent to the usual one. Then, inequalities (5.4) give a precise form of the equivalence relations C1 kek2 ≤ kek2X ≤ C2 kek2 . (5.5) For a solution U (x, y, τ ), we consider the function f (x) = kU ek2X = (U ∗ (x, y, τ )X(x)U (x, y, τ )e, e) . Differentiating and using (5.2), we get ∂f ∂x U ∗ (Aτ + B)∗ X + X(Aτ + B) + X 0 U e, e = −τ (U e, U e) + B ∗ X + XB + X 0 U e, U e . = The matrix B ∗ X + XB + X 0 is Hermitian and, for τ large enough, we have τ τ − ≤ B ∗ X + XB + X 0 ≤ 2 2 in the sense of quadratic forms. By (5.5), the norm kU ek2 may be replaced by kU ek2X , hence ∂f ≤ −d τ f (x) ∂x 42 B. FEDOSOV, B.-W. SCHULZE, AND N. TARKHANOV with some positive constant d. Dividing by f (x) and integrating from y to x, with x ≥ y, we obtain log f (x) ≤ −d τ (x − y) f (y) which means that kU (x, y, τ )ek2X(x) ≤ exp(−d (x − y) τ ) kek2X(y) . Since the norms k · kX(x) are equivalent to any fixed norm k · k, we come to (5.1), which completes the proof. This theorem has some obvious modifications. For example, an estimate (5.6) kU (x, τ )U −1 (y, τ )k ≤ C exp(−d (x − y) τ ) holds if τ → −∞ and x ≤ y. Next, we may replace a stable matrix A = a− by a matrix A = a+ with a spectrum in the right half-plane. In this case we have (5.7) kU (x, τ )U −1 (y, τ )k ≤ C exp(d (x − y) τ ) for τ → +∞ and x ≤ y or τ → −∞ and x ≥ y, with some C, d > 0. 6. Asymptotics of solutions. In this section we consider the general case of Equation (4.1) with a split matrix A(x). So, we write it in the form ∂U = (Λ(x, τ ) + B(x)) U ∂x (6.1) where λ+ (x, τ ) 0 0 λ− (x, τ ) a+ (x)τ + b11 (x) 0 0 a− (x)τ + b22 (x) Λ(x, τ ) = (6.2) = is a block-diagonal part and B(x) = 0 b12 (x) b21 (x) 0 is an antidiagonal part of the coefficients. We assume that both a− (x) and −a+ (x) are stable matrices. Let us look for a solution of (6.1) in the form (cf. (4.5) in [FT87, Ch. 1]) (6.3) U (x, τ ) = (1 + W (x, τ ))Z(x, τ ), INDEX FORMULA FOR SINGULAR SURFACES 43 where Z is a block-diagonal matrix and W is an antidiagonal matrix. Substituting (6.3) into (6.1) and separating diagonal and antidiagonal parts, we obtain ∂W ∂Z Z +W = ΛW Z + BZ, ∂x ∂x (6.4) ∂Z = (Λ + BW ) Z. ∂x Eliminating Z, we arrive at a matrix Riccati equation for W ∂W = ΛW − W Λ + B − W BW. ∂x Were W a solution of (6.5), the second equation in (6.4) would give us an equation for Z with a block-diagonal coefficient Λ + BW . To find W , we observe that Equation (6.5) is equivalent to two separate equations for w12 and w21 , (6.5) ∂w12 = λ+ w12 − w12 λ− + b12 − w12 b21 w12 , ∂x ∂w21 (6.7) = λ− w21 − w21 λ+ + b21 − w21 b12 w21 . ∂x Assuming λ± to be of the form (6.2), let us consider τ positive and large enough. We will look for solutions to (6.6) and (6.7) on the closed interval x ∈ [0, 2π] with initial conditions (6.6) (6.8) (6.9) w12 (2π) = 0, w21 (0) = 0. Lemma 6.1. The solutions of (6.6), (6.8) and (6.7), (6.9) exist, for τ large enough, and satisfy the estimates 1 , kw12 (x, τ )k = O τ (6.10) 1 kw21 (x, τ )k = O τ uniformly in x ∈ [0, 2π]. Proof. Let us consider the case of w12 , the reasoning for w21 is similar. First we reduce (6.6), (6.8) to an equivalent integral equation. To this end, let us treat f = b12 − w12 b21 w12 as a known function and apply the variation of constants to the equation 0 w12 = λ+ w12 − w12 λ− + f. In other words, we look for a solution of the form (6.11) w12 (x) = U+ (x)V (x)U−−1 (x) 44 B. FEDOSOV, B.-W. SCHULZE, AND N. TARKHANOV where U± (x, τ ) are fundamental solutions to the Cauchy problems ∂U± = λ± U± , ∂x U± |x=0 = 1. Substituting, we obtain ∂V = U+−1 f U− ∂x and taking into account (6.8), Z V (x) = − 2π U+−1 (y) f (y) U− (y) dy. x Now, returning to (6.11) and replacing f (y), we come to the integral equation Z 2π w12 (x) = − U+ (x)U+−1 (y) (b12 (y) − w12 (y)b21 (y)w12 (y)) U− (y)U−−1 (x)dy. x This equation may be solved by iterations. From Theorem 5.1 and what has been said at the end of Section 5, we deduce that (6.12) kU+ (x)U+−1 (y)k ≤ C exp(d (x − y) τ ), (6.13) kU− (y)U−−1 (x)k ≤ C exp(d (x − y) τ ) for τ 1 and x ≤ y. In particular, these expressions are uniformly bounded for τ 1 and 0 ≤ x ≤ y ≤ 2π. The initial iteration Z 2π − U+ (x)U+−1 (y) b12 (y) U− (y)U−−1 (x) dy x may be estimated by means of (6.12), (6.13) as Z 2π C exp(2 d (x − y) τ ) dy ≤ x C 2dτ 1 = O . τ When combined with the boundedness of (6.12) and (6.13), this estimate implies the convergence of the iterations and the desired estimate (6.10). Similarly, for w21 we obtain an integral equation Z x w21 (x) = U− (x)U−−1 (y) (b21 (y) − w21 (y)b12 (y)w21 (y)) U+ (y)U+−1 (x)dy 0 and then repeat the previous arguments. INDEX FORMULA FOR SINGULAR SURFACES 45 Turning to the block-diagonal part, we denote by Z± (x, τ ) the entries of Z. More precisely, we take them as solutions of the Cauchy problems ∂Z+ = (λ+ + b12 w21 ) Z+ , ∂x Z+ |x=0 = 1 (6.14) and ∂Z− = (λ− + b21 w12 ) Z− , ∂x Z− |x=0 = 1. (6.15) The crucial property of the coefficients in (6.14) and (6.15) is that, for τ 1, the matrix 1 λ− + b21 w12 = λ− + O τ is stable and so is − (λ+ + b12 w21 ) . In particular, this implies estimates (5.1), (5.6) for Z− and (5.7) for Z+ . We have thus constructed a solution of the form (6.3), with 1 W (x, τ ) = O τ uniformly in x. It does not satisfy the initial condition U (0, τ ) = 1, but this drawback can be easily corrected. Indeed, V (x, τ ) = U (x, τ )U −1 (0, τ ) = (1 + W (x, τ ))Z(x, τ )(1 + W (0, τ ))−1 is the desired solution. For the monodromy matrix, we obtain (6.16) M (τ ) = V (2π, τ ) = (1 + W (2π, τ ))Z(2π, τ )(1 + W (0, τ ))−1 1 1 Z+ (2π, τ ) 0 = 1+O 1+O . 0 Z− (2π, τ ) τ τ Finally, we apply (6.16) to compute the asymptotic expansion of the phase function 1 ϕ(τ ) = log det M (τ ) + M −1 (τ ) − 2 2 46 B. FEDOSOV, B.-W. SCHULZE, AND N. TARKHANOV for <τ → ±∞ and |=τ | ≤ C. All the calculations will be performed modulo πi. From (6.16) it follows that 1 1 Z(2π, τ ) 1 + O M (τ ) + M −1 (τ ) − 2 = 1 + O τ τ 1 1 1 −1 Z (2π, τ ) − 2 1 + O 1+O + 1+O τ τ τ implying 1 1 ϕ(τ ) = log det Z(2π, τ ) 1 + O 2 τ 1 1 1 −1 + 1+O Z (2π, τ ) − 2 1 + O +O . τ τ τ A straightforward computation shows that the expression in curly brackets transforms further to 1 1 0 Z+ (2π, τ ) 0 1 0 1+O 0 Z− (2π, τ ) 0 1 0 Z− (2π, τ ) τ −1 −1 1 Z+ (2π, τ ) 0 Z+ (2π, τ ) 0 + 1+O 0 1 0 1 τ −1 1 1 0 1 0 Z+ (2π, τ ) 0 −2 1+O . −1 0 Z− (2π, τ ) 0 1 0 Z− (2π, τ ) τ −1 Now, Z+ (2π, τ ) and Z− (2π, τ ) decay exponentially for <τ → +∞. Indeed, applying (5.1) for x = 2π and y = 0, we get kZ− (2π, τ )k ≤ C exp(−2πdτ ) 1 = O ; τ −1 the same is true for Z+ (2π, τ ), as may be seen from (5.7) for x = 0 and y = 2π. Hence the previous expression can be rewritten as 1 1 0 Z+ (2π, τ ) 0 1+O −1 0 1 0 Z− (2π, τ ) τ so that 1 1 ϕ(τ ) = log det Z+ (2π, τ ) − log det Z− (2π, τ ) + O 2 2 1 . τ INDEX FORMULA FOR SINGULAR SURFACES 47 Finally, using the Liouville formula for detZ+ and detZ− , we arrive at (6.17) Z 2π Z 2π 1 1 1 ϕ(τ ) = tr (λ+ + b12 w21 )dx − tr (λ− + b21 w12 )dx + O 2 0 2 0 τ Z 2π 1 1 (tr (a+ (x)τ + b11 (x)) − tr (a− (x)τ + b22 (x))) dx + O . = 2 0 τ Similarly, an asymptotic formula for ϕ(τ ) may be obtained as <τ → −∞ and |=τ | ≤ C. The result will be given by (6.17) with the opposite sign. We summarize these results as follows. Theorem 6.2. Let <τ → ±∞ and |=τ | ≤ C. Then the following asymptotic formulas hold: (6.18) Z 2π 1 (tr (a+ (x)τ + b11 (x)) − tr (a− (x)τ + b22 (x))) dx + πiN± ϕ(τ ) = ± 2 0 1 +O . τ The integers N± remain undetermined. We may fix one of them, then the other will depend on the path to be used for analytic extension. Corollary 6.3. The variation ∆Γ,τ0 ϕ(τ ) is an integer multiple of πi. Proof. Using (6.18) we write ∆Γ,τ0 ϕ(τ ) = πi (N+ − N− ) Z 1 2π (tr (a+ (x)(τ0 + T ) + b11 (x)) − tr (a− (x)(τ0 + T ) + b22 (x))) dx + 2 0 Z 1 2π (tr (a+ (x)(τ0 − T ) + b11 (x)) − tr (a− (x)(τ0 − T ) + b22 (x))) dx. + 2 0 The two integral terms give Z 2π (tr (a+ (x)τ0 + b11 (x)) − tr (a− (x)τ0 + b22 (x))) dx 0 which is zero in virtue of Lemma 4.1. References [APS75] M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry, I, Math. Proc. Camb. Phil. Soc., 77 (1975), 43-69. [BFK91] D. Burghelea, L. Friedlander and T. Kappeler, On the determinant of elliptic differential and finite difference operators in vector bundles over S1 , Commun. Math. Phys., 138 (1991), 1-18. 48 [FT87] B. FEDOSOV, B.-W. SCHULZE, AND N. TARKHANOV L.D. Faddeev and L.A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer-Verlag, Berlin et al., 1987. [FST97] B.V. Fedosov, B.-W. Schulze and N.N. Tarkhanov, The Index of Elliptic Operators on Manifolds with Conical Points, Preprint 97/24, Univ. Potsdam, Potsdam, August 1997. [FST97] , On the Index Formula for Singular Surfaces, Preprint 97/31, Univ. Potsdam, Potsdam, November 1997. [Gan86] F.R. Gantmacher, The Theory of Matrices, Springer-Verlag, Berlin et al., 1986. [GS83] P.B. Gilkey and L. Smith, The twisted index problem for manifolds with boundary, J. Differential Geom., 18 (1983), 393-444. [Mel93] R.B. Melrose, The Atiyah-Patodi-Singer Index Theorem, A K Peters, Wellesley, Mass., 1993. [Mel95] , The eta invariant and families of pseudodifferential operators, Math. Res. Letters, 2(5) (1995), 541-561. [SSS97] B.-W. Schulze, B. Sternin and V. Shatalov, On the Index of Differential Operators on Manifolds with Conical Singularities, Preprint 97/10, Univ. Potsdam, Potsdam, May 1997. Received March 1, 1998. The third author was supported by the Max-Planck Gesellschaft. Universität Potsdam 14415 Potsdam Germany Universität Potsdam 14415 Potsdam Germany Universität Potsdam 14415 Potsdam Germany E-mail address: [email protected] PACIFIC JOURNAL OF MATHEMATICS Vol. 191, No. 1, 1999 ORDERING THE BRAID GROUPS R. Fenn, M.T. Greene, D. Rolfsen, C. Rourke, and B. Wiest We give an explicit geometric argument that Artin’s braid group Bn is right-orderable. The construction is elementary, natural, and leads to a new, effectively computable, canonical form for braids which we call left-consistent canonical form. The left-consistent form of a braid which is positive (respectively negative) in our order has consistently positive (respectively negative) exponent in the smallest braid generator which occurs. It follows that our ordering is identical to that of Dehornoy (1995) constructed by very different means, and we recover Dehornoy’s main theorem that any braid can be put into such a form using either positive or negative exponent in the smallest generator but not both. Our definition of order is strongly connected with Mosher’s (1995) normal form and this leads to an algorithm to decide whether a given braid is positive, trivial, or negative which is quadratic in the length of the braid word. 0. Introduction. Dehornoy [5, 6, 7] has proved that the braid group is right-orderable. More precisely, there is a total order on the elements of the braid group Bn which is right invariant in the following sense. Suppose that α, β, γ ∈ Bn and α < β, then αγ < βγ. This ordering is uniquely defined by the condition that a braid β0 σi β1 is positive (i.e. greater than the identity braid), where ±1 ±1 β0 , β1 are words in σi+1 , . . . , σn−1 . Dehornoy’s proof is based on some highly complicated algebra connected with left-distributive systems. In this paper we construct this order geometrically using elementary arguments. Our construction leads to a new, effectively computable, canonical form for braids which we call left-consistent canonical form. The left-consistent form of a positive braid has the general shape β0 σie β1 . . . βl−1 σie βl where the βi are words in σi+1 , . . . , σn−1 and their inverses, and e = +1. For a negative braid the form is similar but with e = −1. It follows at once that our ordering is identical to Dehornoy’s and we recover Dehornoy’s main theorem that any braid can be put into such a shape for e = 1 or e = −1 but not both. 49 50 FENN, GREENE, ROLFSEN, ROURKE, AND WIEST Our definition of order is strongly connected with Mosher’s automatic structure [13] and this implies that the braid group is order automatic, i.e., the order can be detected from the automatic normal form by a finite state automaton. Furthermore the resulting algorithm to decide whether a given braid is positive, trivial, or negative is linear in the length of the Mosher normal form and hence quadratic in the length of the braid word (in contrast, Dehornoy’s algorithm [7], although apparently fast in practice is only known to be exponential). The paper is organised as follows. Section 1 contains basic definitions and introduces the curve diagram associated to a braid. In Section 2 we prove that a curve diagram can be placed in a unique reduced form with respect to another and in Section 3 we define the order by comparing the two curve diagrams in reduced form, and prove that it is right-invariant. In Section 4 we construct the left-consistent canonical form of a braid, deduce that our order coincides with Dehornoy’s and recover Dehornoy’s results. In Section 5 we give some counterexamples connected with the order, and in Section 6 we make the connection with Mosher’s normal form and deduce the existence of the quadratic time algorithm to detect order. Finally, in an appendix, we use cutting sequences to give a formal algorithm to turn a braid into the new left-consistent canonical form; note that this algorithm is not quadratic time. Acknowledgements. We are grateful to the organisers of the low-dimensional topology conference held at the Isle of Thorns in Spring 1997, which was supported by the LMS, for providing a congenial atmosphere for the initial work on this paper. We are also grateful to Caroline Series for suggesting that our curve diagrams might be related to Mosher’s normal form for mapping class groups. We would also like to thank the referee for helpful comments and a speedy report. Bert Wiest is supported by a TMR (Marie Curie) research training grant. 1. Braids and curve diagrams. Let D2 be the closed unit disk in C, and let Dn be the disk D2 with n distinct points in the real interval (−1, 1) removed. We consider the group Bn of self-homeomorphisms γ : Dn → Dn with γ|∂Dn = id, up to isotopy of Dn fixed on ∂Dn . Multiplication in Bn is defined by composition. The group Bn is well-defined independently of the points removed; indeed if Dn0 is a disk with any n-tuple of points removed and Bn0 the corresponding group then there is an isomorphism Bn0 ∼ = Bn ; if these points also lie on (−1, 1) then this isomorphism is natural. en of braids on n strings, with The group Bn is isomorphic to the group B multiplication given by concatenation: If α, β are braids (pictured vertically) ORDERING THE BRAID GROUPS 51 then α · β is α above β. It is well known that this group has presentation en ∼ B = hσ1 , . . . , σn−1 | σi σj = σj σi if |i − j| > 2, σi+1 σi σi+1 = σi σi+1 σi i, where the generator σi (i ∈ {1, . . . n − 1}) is indicated in Figure 1. 1 2 i−1 i i+1 i+2 ... n−1 n ... Figure 1. The standard generator σi of the braid group on n strings. en → Bn is given by ‘putting the braid in a solid The isomorphism B cylinder and sliding Dn once along it’. The inverse map is defined as follows: Extend a given homeomorphism γ : Dn → Dn to a homeomorphism γ 0 : D2 → D2 , then find a boundary-fixing isotopy γt : D2 → D2 with γ0 = id and γ1 = γ. Then the flow of the n holes of Dn under γt defines a braid on n strings. For details see [2]. On Dn we draw n + 1 line segments as in Figure 2(a). If γ is a homeomorphism of Dn representing an element [γ] of Bn , then γ sends these line segments to n + 1 disjoint embedded curves, and if [γ1 ] = [γ2 ] ∈ Bn then γ1 and γ2 give rise to isotopic collections of curves. For instance, Figure 2 shows the effect of the braid σ1 σ2−1 ∈ B3 . Here the holes of Dn , as well as ±1 are indicated by black dots. We call such a diagram of n + 1 disjoint simple curves in an n–punctured disk a curve diagram, and we number the curves in the diagram 1 to n + 1, as in Figure 2. (a) (b) (c) 4 1 1 2 3 4 ·σ1 2 4 3 ·σ2−1 1 2 3 Figure 2. Curve diagrams of the braids 1, σ1 , and σ1 σ2−1 . Conversely, from the curve diagram we can reconstruct the homeomorphism γ up to boundary-fixing isotopy; that is, we can reconstruct the element of the braid group. 52 FENN, GREENE, ROLFSEN, ROURKE, AND WIEST 2. Reduced form. Let Γ and ∆ be curve diagrams of two braids γ and δ, say. In order to compare Γ and ∆, we superimpose the two diagrams and reduce the situation by removing unnecessary intersections. This process is well known and often called “pulling tight” (see e.g. [13]). We will denote the ith curve of a curve diagram such as Γ by Γi . The curves Γi and the ∆j are called parallel if they connect the same pairs of points and are isotopic in Dn . For instance, curve 3 of Figure 2(b) and curve 2 of Figure 2(c) are parallel. We define ∆ to be transverse to Γ if every curve of ∆ either coincides precisely with some (parallel) curve of Γ, or intersects the curves of Γ transversely. We define the intersection index of two transverse curve diagrams to be n + 1 + #(transverse intersections) − #(coincident curves). (The geometric significance is that Dn cut along Γ has two components, and cutting in addition along ∆ increases the number of components by the intersection index.) For example, the diagrams in Figure 2(a) and 2(c) have intersection index 6, the diagrams in Figure 2(a) and 2(b) have intersection index 2 and the diagrams in Figure 2(b) and 2(c) have intersection index 5. The intersection index of two curve diagrams is 0 if and only if the diagrams are identical. We now fix a curve-diagram Γ for γ, and look at all possible curvediagrams for δ. They are all isotopic in Dn , but they may have very different intersection-indices with Γ. We say ∆ and ∆0 are equivalent (with respect to Γ) if they are related by an isotopy of Dn , which is fixed on curves of ∆ which coincide with curves of Γ, and which leaves the diagrams transverse all the time. So coincident curves remain coincident, and the intersection index remains unchanged. We define a D-disk1 between ∆ and Γ to be a subset of Dn homeomorphic to an open disk, which is bounded by one open segment of some curve of ∆, one open segment of some curve of Γ, and two points, each of which may be an intersection-point of the two curves or one of the ‘holes’ of Dn , or ±1 ∈ Dn . There are three types of D-disks (types (a), (b), and (c)), indicated in Figure 3, where the curve-diagram Γ is drawn with dashed, and ∆ with solid lines, and the dots denote holes or ±1. If there are no D-disks between ∆ and Γ then we say ∆ and Γ are reduced. If the curve-diagrams ∆ and Γ are not reduced, i.e. if they have a D-disk, then we can isotope ∆ so as to reduce the intersection index with Γ (Figure 3). This isotopy consists of ‘sliding a segment of a curve of ∆ across the D-disk’ (for reduction moves (I) and (II)), and of ‘squashing a D-disk to a line’ (for reduction move (III)). The three moves reduce the intersection 1 D-disks are often called “bigons” in the literature. ORDERING THE BRAID GROUPS (a) 53 (I) (b) (II) (c) (III) Figure 3. Three types of D–disks, and how to use them to reduce intersection indices. index by 2, 1, 1, respectively. We observe that any curve-diagram ∆ with intersection index 0 with Γ is reduced. Thus we can reduce curve diagrams by a finite sequence of ‘isotopies across D-disks’ as in Figure 3. Lemma 2.1 (Triple reduction lemma). Suppose Σ, Γ and ∆ are three curve diagrams such that Γ and ∆ are both reduced with respect to Σ. Then there exists an isotopy between ∆ and a curve diagram ∆0 , which is an equivalence with respect to Σ, such that Σ, Γ and ∆0 are pairwise reduced. Proof. We consider a D-disk bounded by one segment of curve of Γ and one of ∆. This D-disk may have several intersections with Σ. There are, a priori, three possibilities for the type of such an intersection — they are indicated in Figure 4, labelled (1), (2), and (3). (In this figure, the D-disk is of type (b), the cases of types (a) and (c) are similar.) However, (1) and (2) are impossible, because Γ and ∆ are reduced with respect to Σ. So all intersections are of type (3), and the D-disk can be removed without disturbing the reduction of Σ with respect to Γ or ∆, as indicated in Figure 4. The statement follows inductively. (3) (1) (II) (2) Figure 4. The solid line is ∆, the dashed Γ, and the dotted Σ. Lemma 2.2. If two isotopic curve diagrams Γ and ∆ are reduced with respect to each other, then they coincide. 54 FENN, GREENE, ROLFSEN, ROURKE, AND WIEST Proof. Suppose that the first curve Γ1 of Γ does not coincide with the first curve ∆1 of ∆. Consider the word obtained by reading the intersections of Γ1 with the curves of ∆ in order. Since Γ1 is isotopic to ∆1 , this word must cancel to the trivial word. It follows by a simple innermost disk argument that there must be a D-disk. Hence Γ1 must coincide with ∆1 . Similarly all curves of Γ and ∆ must coincide. Proposition 2.3. If two curve diagrams ∆ and ∆0 of a braid δ are reduced with respect to Γ then they are equivalent with respect to Γ. Proof. By the triple reduction lemma we may reduce ∆ with respect to ∆0 by an isotopy of ∆ which is an equivalence with respect to Γ. After this reduction ∆ and ∆0 coincide by Lemma 2.2. We have proved that by reducing a curve diagram ∆ with respect to a curve diagram Γ we can bring ∆ into a uniquely defined standard form with respect to Γ. In particular reduction of ∆ with respect to the trivial curve diagram representing 1 ∈ Bn (Figure 2(a)) leads to a canonical representation of braids in terms of cutting sequences, which will be discussed in detail in the appendix. Remark 2.4. The following observation will be crucial at a later point. Let Γ and ∆ be transverse curve diagrams. Suppose the curve ∆i on its own is reduced with respect to Γ. Then we can reduce ∆ with respect to Γ by an isotopy of ∆ which is fixed on ∆i . 3. The right-invariant order on Bn . We define a total ordering on the braid group Bn as follows. Suppose γ and δ are two braids on n strings. We let Γ be a curve diagram for γ, and ∆ be a curve diagram for δ which is reduced with respect to Γ. The collection of curves of Γ cuts Dn into two components, √ which we call the √ upper and the lower component, containing the points −1 respectively − −1 in Dn ⊆ C. We orient the curves of ∆ coherently such that we obtain a path starting at −1 ∈ Dn and ending at 1 ∈ Dn . If all curves of ∆ coincide with the corresponding curves of Γ then the braids γ and δ are equal. Suppose that curves 1, 2, . . . , i − 1 of ∆ agree with the corresponding curves of Γ, and the ith is the first transverse one, 1 ≤ i ≤ n + 1. This oriented curve has the same startpoint as the ith curve of Γ, and first branches off Γ either into the upper or the lower component. In the first case we define δ > γ, in the second δ < γ. This is well-defined, by Proposition 2.3. Example. All the diagrams in Figure 2 are reduced with respect to each other, and we observe that 1 < σ1 · σ2−1 < σ1 . ORDERING THE BRAID GROUPS 55 Proposition 3.1. The relation ‘<’ is an ordering, i.e., if σ, γ, δ are braids with σ < γ < δ then σ < δ. Proof. By the triple reduction Lemma 2.1 we can find curve diagrams Σ, Γ, ∆ of these braids which are all pairwise reduced. The statement of the proposition follows immediately: If Γ branches off Σ to the left and ∆ branches off Γ to the left, then ∆ branches off Σ to the left. Proposition 3.2. The ordering ‘<’ is right invariant. Proof. Suppose we have two braids δ and γ with δ < γ, and with reduced curve diagrams ∆ and Γ. Let σ be a further braid, i.e. a homeomorphism of Dn fixing ∂Dn . We obtain the curve diagrams for δ · σ and γ · σ by applying σ to ∆ and Γ. The resulting curve diagrams σ(∆) and σ(Γ) are still reduced, and σ(∆) still branches off σ(Γ) into the lower component of Dn \ σ(Γ), so δ · σ < γ · σ. Let be the trivial braid, with standard curve diagram E (see Figure 2(a)). We call a braid γ positive if γ > , and negative if γ < . If we want to stress that the first i − 1 curves of Γ are parallel to the corresponding curves of E, and the ith is the first non-parallel one, then we say γ is i-positive respectively i-negative. Since there is a very similar concept of σi -positive (see the next section) we shall often say geometrically i-positive or negative. Given two braids γ and δ such that γ > δ we say γ is (geometrically) igreater than δ if the ith curves are the first non-parallel ones. Any curve diagram in which the first i − 1 curves are parallel to the corresponding curves of E is called (i − 1)-neutral. We note some simple consequences of right invariance. We have γ > if and only if > γ −1 , so the inverse of a positive braid is negative. If γ > and δ is any braid, then γδ > δ. (Warning: it need not be true that δγ > δ — see the next section.) In particular, the product of positive braids is positive. 4. Left-consistent canonical form. In this section we connect our ordering with Dehornoy’s [5]. The following definition is taken from [5]. A word of the form β0 σi β1 σi . . . σi βk , ±1 ±1 where i ∈ {1, . . . , n − 1}, and β0 , . . . βk are words in the letters σi+1 . . . σn−1 is called a σi -positive word. A braid is σi -positive if it can be represented by a σi -positive word. A braid is called σi -negative if its inverse is σi -positive. We shall say that a braid is σ-positive or negative if it is σi -positive or negative for some i. The following is the main result from [5]: Dehornoy’s theorem. Every braid is precisely one of the following three: σ-positive, or σ-negative, or trivial. 56 FENN, GREENE, ROLFSEN, ROURKE, AND WIEST Dehornoy uses this theorem to define a right-invariant order by α > β ⇐⇒ αβ −1 is σ-positive. We shall prove that this order coincides with the order we defined in the last section by showing that the concepts of geometrically i-positive and σi -positive coincide. One way is easy. Proposition 4.1. A braid which is σi -positive is geometrically i-positive. Proof. A braid which can be represented by a word βσi β 0 , where β, β 0 are words in the letters σi+1 , . . . , σn−1 , is geometrically i-positive. To see this think of the homeomorphism determined by the braid word as a sequence of twists of adjacent holes around each other: β leaves the first i curves untouched and then σi twists the ith hole around the (i + 1)st producing a curve diagram in which the ith curve moves into the upper half of Dn ; finally β 0 leaves the start of the i-th curve untouched. Now by definition, every Dehornoy positive braid is a product of such words. The proposition now follows from the fact that the product of two geometrically i-positive braids is again geometrically i-positive. The proposition immediately implies part of Dehornoy’s theorem: every braid can take at most one of the three possible forms. To complete the proof that the concepts of geometrically i-positive and σi -positive coincide and to recover the remainder of Dehornoy’s theorem we shall construct a canonical σi -positive form for a given geometrically i-positive braid. This is the left-consistent canonical form of the braid: Theorem 4.2 (Left-consistent canonical form). Let γ be a geometrically ipositive braid. Then there is a canonically defined σi -positive word which represents the same element of Bn . We define the complexity of a braid γ as follows. Take a curve diagram Γ for γ which is reduced with respect to the trivial curve diagram E. Suppose that the first j − 1 curves of Γ coincide with the first j − 1 curves of E and that the jth curve does not. Let m ≥ 0 be the number of transverse intersections of Γ with jth curve of E. The complexity of γ is the pair (j, m). We order complexity lexicographically with j in reverse order. Thus (1, m) is more complex than (2, n) for any m, n whilst (j, m) is more complex than (j, n) if and only if m > n. The main step in the proof of Theorem 4.2 is the following: Proposition 4.3. Suppose that γ is a geometrically i-positive braid. Then there is a word β in the braid generators σi , . . . , σn−1 and their inverses such that (1) β contains σi−1 exactly once (2) β does not contain σi (3) γβ is either geometrically i-positive or geometrically i-neutral (4) γβ has smaller complexity than γ. ORDERING THE BRAID GROUPS 57 Furthermore there is a canonical choice for β. Theorem 4.2 follows from Proposition 4.3 by induction on complexity because, by (4) and induction, γ 0 := γβ has a canonical form which by (3) is either σi -positive or σj -positive or negative for j > i and then γ 0 β −1 is the canonical form for γ. Proof of Proposition 4.3. For definiteness we shall deal with the case i = 1 first. (We shall see that the general case is essentially the same as this case.) So let γ be a geometrically 1-positive braid and Γ a curve diagram for γ which is reduced with respect to the trivial curve diagram. We shall define β geometrically by sliding one particular hole of Dn along a useful arc. Let E1 ⊆ Dn be the 1st curve of E, i.e. a straight line from −1 to the leftmost hole of Dn , excluding this hole. We define a useful arc to be a segment b of some curve of Γ starting at some point of E1 (possibly −1), and ending at some hole of Dn other than the leftmost one such that - the interior of b does not intersect E1 , - an initial segment of the arc b lies in the upper half of the disk, i.e. the intersection of a neighbourhood of E1 with the interior of b consists of a line segment in the upper component of Dn \ E. (a) (b) Figure 5. Slide of a hole along a useful arc, followed by a reduction. Suppose that Γ contains useful arcs. Then each of them has precisely one point of intersection with E1 and we call the one whose intersection point is leftmost the leftmost useful arc. Let b be the leftmost useful arc. If b starts in the interior of E1 , then we can slide the hole of Dn at the endpoint of 58 FENN, GREENE, ROLFSEN, ROURKE, AND WIEST b along b and back into E1 . If b starts at −1, then we push a small initial segment of b into E1 , and then perform the slide of the hole of Dn (see Figure 5, where b is dotted). In either case we obtain a curve diagram Γ0 representing a braid γ 0 . Now Γ0 need not be reduced with respect to E. But notice that γ 0 has lower complexity than γ since the new E1 now stops at the intersection of b with the old E1 and hence there are fewer intersections with Γ0 even before reduction. The movement of the hole of Dn along b defines a braid β on n strings, with γ 0 = γβ. Furthermore we can decompose β as a canonical word in the generators σj by writing down the appropriate σj or σj−1 whenever the hole passes over or under another hole. But, by definition of useful arc, the hole only passes once over or under the leftmost hole and it passes over and to the left and hence the word that we read contains σ1−1 only once and does not contain σ1 . Therefore to prove case i = 1 of Proposition 4.3 it remains to prove the following two claims: Claim 1. The diagram contains a useful arc. Claim 2. The diagram Γ0 obtained by sliding a hole of Dn along the leftmost useful arc is either 1-positive or 1-neutral, but not 1-negative. To prove Claim 1, we consider the first curve of Γ starting at −1. If it ends in a hole other than the leftmost one and does not intersect E1 then it is a useful arc (Figure 6(a)). Otherwise we consider the closed curve in D2 starting at −1, along the first curve of Γ, up to its first intersection with the closure of E1 in D2 , and then back in a straight line to the point −1. This curve bounds a disk S in D2 , which may be of three different types: Γ hits E1 either from above, or from below, or in the leftmost hole of Dn (see Figure 6(b),(d),(c)). In cases (b) and (c) we note that since Γ and E are reduced, at least one hole of Dn must lie in the interior of S. Moreover, all holes of Dn are connected by curves of Γ, so there exists a curve of Γ connecting one of the holes in S to one of the holes outside S or the point 1 ∈ Dn . The first component of the intersection of this curve with S is a useful arc. In case (d) we walk along the oriented curve in D2 starting at −1, along the curves of Γ. We write down the symbol + whenever we hit E1 from below and − if we hit E1 from above or in the leftmost hole of Dn . The sequence starts with a +, and since the curve has to leave the disk S it must contain a −. It follows that the string +− must occur in the sequence; it represents an arc which, together with a segment of E1 , bounds a disk S 0 in D2 . See Figure 6(d): S 0 is bounded by part of the dotted arc between two intersections with E1 and part of E1 . Since Γ and E are reduced, S 0 contains a hole other than the leftmost one in its boundary or in its interior. In the first case, a segment of top (dotted) boundary of S 0 is a useful arc; in ORDERING THE BRAID GROUPS (a) 59 (b) S E1 (c) (d) S S S ′ Figure 6. How to find a useful arc. the second case the disk S 0 is of the type indicated in Figure 6(b) or (c), so there is a useful arc inside S 0 . This finishes the proof of Claim 1. To prove Claim 2, we distinguish two cases: Either the leftmost useful arc b starts at the point −1, or it starts at some point in the interior of E1 . In the first case (e.g. Figure 5(b)), the curve diagram Γ0 obtained by sliding a hole along b to near −1 is 1–neutral. In the second case (Figure 5(a)) the curve diagram Γ0 is 1-positive, as we now prove. We recall that we had γ 0 = γβ, where β represents the slide of a hole along the leftmost useful arc b. We observe that we can construct a curve diagram of the braid β −1 such that the first curve b1 of the diagram is a line segment in E1 from −1 almost all the way to E1 ∩ b, followed by an arc parallel and close to the arc b, and finally running into the same hole as b. The construction of the arc b1 is illustrated in Figure 7(a). Next we examine the possible reductions of Γ with respect to this arc b1 . If there was a D-disk of type (b) whose boundary contained the arc b, (i.e. to the right of b1 in Figure 7) then cutting off the strip bounded by b, b1 and E1 would yield a D-disk of type (a) of Γ with respect to E (see Figure 7(a)). This is impossible by hypothesis. If there was a D-disk of type (b) whose boundary contained a final segment of the arc b1 and a segment other than b of a curve of Γ, (i.e. to the left of b1 in Figure 7) then this segment would be a useful arc intersecting E1 more to the left than b (Figure 7(b)), which is also impossible. Finally, any D-disk of type (a) of Γ with respect to b1 would also be a D-disk of Γ with respect to E1 . So there are no D-disks between b1 and Γ. By Remark 2.4 it follows that we can reduce the curve diagram of β −1 with respect to Γ without touching its first curve b1 . We can now observe that γ is 1-greater than β −1 , ie γ 0 is 1-positive, as claimed. This completes the proof of Claim 2. 60 FENN, GREENE, ROLFSEN, ROURKE, AND WIEST (a) no D–disks Γ b1 b E1 (b) useful arc Γ D–disk b1 b Figure 7. There are no D-disks between b1 and Γ. Finally we turn to the case when i may not be 1. In this case the first i − 1 holes are lined up near −1 on the real axis. The same argument as in the case i = 1, only with the i − 1st hole and the line segment Ei playing the role previously played by −1 and E1 respectively, completes the proof of the general case. The proof of Theorem 4.2 provides an explicit algorithm for converting a braid into its left-consistent canonical form. In the appendix we give a formal version of this algorithm using cutting sequences. Remark. The order on the braid group has the property that inserting a generator σi anywhere in a braid word makes the braid larger. A proof of this fact, in the spirit of this paper, is given in [17]. This property is equivalent to the statement that the order extends the subword order defined by Elrifai and Morton [8] and an algebraic proof has been given by Laver [11]. 5. Counterexamples. We shall call a braid word σ-consistent (Dehornoy in [7] calls it reduced) if it is σ-positive, σ-negative or trivial. We have seen in the previous chapter that every braid has at least one σ-consistent representative. The aim of this chapter is to disprove some plausible-sounding but over-optimistic conjectures about the ordering and about σ-consistent representatives of braids. ORDERING THE BRAID GROUPS 1 2 61 3 c a = = d b Figure 8. Equivalence of a conjugate of a positive pure braid with a visibly negative braid. Left invariance on the pure braid group. Because the pure braid group has an ordering which is simultaneously left and right invariant [15], it would be tempting to think that the geometric ordering is left and right invariant when restricted to the pure braid group. However this is equivalent to saying that a pure positive braid, when conjugated by any pure braid, is again positive and the example in Figure 8 shows this to be false. In B3 , the braid group on three strings, we conjugate the pure positive braid σ12 σ2−2 by the pure braid σ2 σ12 σ2 . The figure shows the equivalence of the resulting braid with the σ-negative braid σ2−1 σ1−1 σ23 σ1−1 σ2 σ1−1 . We are moving first the string segment ab and then the segment cd ‘over’ the braid ‘to the left of the braid’. Simultaneously shortest and σ-consistent representatives. For any element b of the braid group Bn (n > 2), there are two ways to ±1 represent b by a particularly simple word w in the letters σ1±1 ,. . . ,σn−1 . (1) b can be represented by a word which is as short as possible. For instance, we shall see later that the word w1 = σ1 σ2 σ3−1 σ2 σ1−1 is a shortest possible representative of a braid in B4 (see Figure 9(a)). (2) b can be represented by a σ-consistent word. For instance, in the braid word w2 = σ2−1 σ3−1 σ1 σ2−1 σ1 σ3 σ2 , which represents the same element of B4 as w1 , the letter σ1 occurs only with positive exponent, see Figure 9(b). 62 FENN, GREENE, ROLFSEN, ROURKE, AND WIEST (a) 1 2 3 4 (b) Figure 9. The equivalent braids σ1 σ2 σ3−1 σ2 σ1−1 and σ2−1 σ3−1 σ1 σ2−1 σ1 σ3 σ2 . Theorem 5.1. Every element of Bn for n = 2, 3 has a simultaneously shortest and σ-consistent representative. By contrast, there are braids in Bn for n > 4 all of whose σ-consistent representatives have non-minimal length. Proof. The case n = 2 is obvious. The case n = 3 follows from the fact that in B3 Dehornoy’s handle-reduction algorithm [7] never increases the length of a braid word, and hence turns any shortest representative of a given braid into a simultaneously shortest and σ-consistent one. For the case n > 4 it suffices to prove that the braid b := σ1 σ2 σ3−1 σ2 σ1−1 ∈ B4 (Figure 9) has length 5, while every σ-consistent representative has more than five letters. To see that every representative has at least five letters we note that the image of b under the natural homomorphism Bn → Sn , from the braid group into the symmetric group, is the permutation (14). This permutation cannot be written as a product of less than five adjacent transpositions. The result follows. We now assume, for a contradiction, that there exists a five-letter representative which is also σ-consistent. This would be a braid on four strands with the following properties: (i) its image under the natural map B4 → S4 is (14), (ii) it has five crossings (i.e., it is a word with five letters), (iii) if we denote by c(i, j) (i, j ∈ {1, . . . , 4}) the algebraic crossing number of the ith and the jth string, then the braid must satisfy c(1, 2) = 1, c(1, 3) = 1, c(1, 4) = −1, c(2, 3) = 0, c(2, 4) = −1, c(3, 4) = 1, (iv) it may contain the letter σ1 , but not σ1−1 (note that there exists a representative of b in which σ1 occurs only positively, so there can’t exist a consistently negative one). There are only three braids satisfying (i)-(iii), pictured in Figure 10, and we observe that none of them satisfies (iv). It follows that no σ-consistent representative of b with only five crossings exists. Minimal number of occurrences of the main generator. ORDERING THE BRAID GROUPS 1 2 3 4 1 2 3 4 63 1 2 3 4 Figure 10. Three candidates for short σ-consistent representatives of b. We define the main generator of a braid word to be the generator with lowest index occurring in the word. It is tempting to think that sliding holes along leftmost useful arcs, as in the left consistent canonical form, is the most efficient way of reducing the number of intersections between the curve diagram and the line segment E1 . This, however, is wrong: Theorem 5.2. There are braids whose left consistent canonical form does not have the minimal number of occurrences of the main generator among all σ-consistent representatives. Proof. We shall show that the braid ∆3 , where ∆ = σ2 σ1 σ2 , has this property. Note that ∆ is just a half-twist, so ∆2 generates the centre of B3 . We have (σ2 σ1 σ2 )3 = σ2 σ2 σ1 σ2 σ2 σ2 σ1 σ2 σ2 , so the braid can be represented by a σ-consistent word in which the main generator σ1 occurs only twice. However, as is easy to check with the help of Figure 11, the left consistent canonical form of the braid is the word (σ2 σ1 σ2 )3 , which contains the main generator σ1 three times. Local indicability. We are indebted to Stephen P Humphries and Jim Howie for pointing out the following. A group is called locally indicable if every finitely generated subgroup has a nontrivial homomorphism to the integers. It was proved by Burns and Hale [4] that locally indicable groups are right-orderable, but it took almost two decades until G Bergman [1] found an example of a group which is right-orderable but not locally indicable; i.e. the class of locally indicable groups is strictly contained in the class of right-orderable groups. We can now give further examples: 64 FENN, GREENE, ROLFSEN, ROURKE, AND WIEST Figure 11. The left consistent canonical form of ∆3 is σ2 σ1 σ2 σ2 σ1 σ2 σ2 σ1 σ2 . Theorem 5.3. The braid group Bn for n > 5 is right orderable but not locally indicable. Proof. It remains to show that Bn is not locally indicable. The commutator subgroup Bn0 of Bn is finitely generated, and for n > 5 the first and second commutator subgroups coincide: Bn0 = Bn00 (see [10]). It follows that the abelianization of Bn0 is trivial, so Bn0 ⊂ Bn has no nontrivial homomorphism to Z. 6. Automatic ordering. Define a right-invariant ordering to be automatic if it can be determined by a finite-state automaton. In this section we shall see that the ordering on the braid group is automatic. This is proved by comparing the order on the braid group as defined in Section 3 with Mosher’s automatic structure [12, 13]. This comparison gives more. Define a group to be order automatic if it is both automatic [9] and right-orderable and such that there is a finite state automaton which detects the order from the automatic normal forms. To be precise, there exists an automatic structure and a finite state automaton, which, given two normal forms for the automatic structure, will decide which represents the greater group element. Theorem 6.1. The braid group Bn is order automatic. Remark 6.2. The algorithm to decide which of two given normal forms is the greater takes linear time in the length of the normal form. Using results from Epstein et al [9] we deduce: Corollary 6.3. There is a quadratic-time algorithm to decide which of two elements of Bn (presented in terms of standard braid generators) is the greater. Full details of the proof of these results can be found in [16]. Here we shall give a short proof of Theorem 6.1 which yields only a quadratic time ORDERING THE BRAID GROUPS 65 algorithm to order normal forms which is nevertheless sufficient to imply Corollary 6.3. In [12, 13] Mosher constructs normal forms for elements of mapping class groups by combing triangulations (and uses them to prove that mapping class groups are automatic). We shall need to sketch Mosher’s normal form in the special case of the braid group. We define the base triangulation B of Dn to have vertices at the n missing √ points and at the four boundary vertices, ±1 and ± −1. The edges of B comprise the four arcs of ∂Dn joining pairs √ of boundary vertices, n + 1 edges along the real axis and 2n edges joining ± −1 to the real vertices not ±1, see Figure 12. We order and orient the edges as indicated. √ 5 2 1 −1 4 8 10 7 −1 3 6 1 9 √ − −1 Figure 12. The base triangulation. An allowable triangulation of Dn is a triangulation with the same vertex set. We identify two allowable triangulations if they differ by a vertex fixing isotopy. A triangulation class is a set of boundary fixing isomorphism classes of allowable triangulations. I.e. two triangulations are in the same class if they are related by the action of an element of the braid group. We now consider the groupoid G which has for objects the set of triangulation classes of Dn and for morphisms the set of ordered pairs (T, T 0 ) of allowable triangulations, where (T, T 0 ) is identified with (h(T ), h(T 0 )) if h ∈ Bn . The morphism goes from the class of T to the class of T 0 . If T and T 0 are in the same class, then there is a unique boundary fixing isomorphism from T 0 to T up to isotopy, i.e. an element of Bn . This determines an isomorphism between the vertex group of G and the braid group Bn . (Note that for this isomorphism, and for compatibility with Mosher’s conventions, we need to replace the algebraic convention for multiplication in the braid group, described in Section 1, by the opposite functional convention, i.e. ΦΨ := Φ ◦ Ψ. The functional convention is used throughout this section; the algebraic convention is used in all other sections and in the appendix.) 66 FENN, GREENE, ROLFSEN, ROURKE, AND WIEST Combing. We consider a particular type of morphism in G. Definition. Flipping an edge. An edge α adjacent to two triangles δ and δ 0 is removed (to form a square of which α is a diagonal) and then the square is cut back into two triangles by inserting the opposite diagonal. We call this morphism “flipping α” and denote it fα , see Figure 13. α fα Figure 13. Flipping an edge. Every morphism q = (B, T ) in G from the base vertex to another vertex is a product of a canonical sequence of flips. To see this, picture q as given by superimposing B and T , and comb T along B. To be precise, first reduce T with respect to B and then consider edge 1 of B. Suppose that, starting at −1, edge one crosses edge α of T . Flip α. Repeat until there are no more crossings of edge 1 with T . (The fact that this process is finite follows from a simple counting argument: one counts the number of intersections of edge 1 with T , except with the next edge of T which is to be flipped. For more detail here see [13, pages 321-322].) Now do the same for edge 2 starting at the non-boundary vertex and continue in this way, using the ordering and orientation of edges of B indicated in Figure 12, until T has been converted into a copy of B. The Mosher normal form of q is the inverse of the sequence of flips described above.2 Notice that unlike the general case described in [13], q is completely characterised by the sequence of flips, there is no need to carry the labelling of T along the combing sequence. In particular, there is no relabelling morphism required here. (This is because ∂Dn is fixed throughout.) Detecting order from the Mosher normal form. To see the connection with order, consider an element (B, T ) of the vertex group at the class of B. There is an element g ∈ Bn (a homeomorphism of Dn fixing ∂Dn ) unique up to isotopy carrying T to B. Conversely given g ∈ Bn the corresponding triangulation pair is (B, g −1 B). 2 Strictly speaking the Mosher normal form is not this flip sequence, which only defines an asynchronous automatic structure, but is derived from it by clumping flips together into blocks called “Dehn twists”, “partial Dehn twists” and “dead ends” (see [13] pages 342 et. seq.). This technicality does not affect any of the results proved here. We prove that order can be detected in linear time from the flip sequence. Since the clumped flip sequence can be unclumped in linear time, this implies that order can be detected in linear time from the strict Mosher normal form. ORDERING THE BRAID GROUPS 67 We observe that if we comb B along g(B) this is combinatorially identical to combing g −1 (B) along B. We call the sequence of flips defined by this combing the combing sequence of g. (The reverse of the combing sequence is the Mosher normal form of g.) The curve diagram of g is part of the triangulation g(B) namely the edges numbered 1, 4, 7, . . . , 3n + 1. Suppose that g is i-positive, then g can be assumed to fix the first i − 1 of these edges (i.e. 1, 4, . . . , 3i − 5) and then, after reduction, can be assumed to fix the corresponding outlying edges (i.e. 2, 5, . . . , 3i − 4 and 3, 6, . . . , 3i − 3). But edge 3i − 2 is carried into the upper half of Dn and must meet edge 3i−1 of B. Thus the first flip in the combing sequence of g is f3i−1 , i.e. flip the edge numbered 3i − 1. Similarly if g is i-negative then the first flip in the combing sequence is f3i . We have proved the following: Algorithm 6.4. (To decide from the Mosher normal form whether a braid element is i-positive or negative and provide the correct value of i.) Inspect the combing sequence (the reverse of the Mosher normal form). The first flip is either f3i−1 for some i or f3i for some i. In the first case the braid is i-positive and in the second it is i-negative. This algorithm is visibly executable by a finite-state automaton and linear in the length of the normal form of g. Theorem 6.1 and Corollary 6.3 follow from general principles. To decide the relative order of two elements α and β we compute the normal form of αβ −1 — this can be done by a finite-state automaton and takes quadratic time, see [9] — and then apply Algorithm 6.4. Final remarks. (1) We have proved that there is a quadratic time algorithm to decide the relative order of two braid words. In [7] Dehornoy presents an algorithm which does this in practice and is apparently extremely fast — however his formal proof that this algorithm works only provides an exponential bound on time. The algorithm presented here is implementable since the whole Mosher program can be implemented, see [14]. Note that in the appendix we present another algorithm based on cutting sequences. (2) There is a far stronger connection between the Mosher normal form and the order on Bn than presented here. The relative order of two elements can be detected from their combing sequences by inspecting just the first four differences in the sequences (and this proves Remark 6.2). Full details here are to be found in [16]. Appendix A. Cutting sequences. In this appendix we define a unique reduced cutting sequence for a braid. We give implementable algorithms to read the reduced cutting sequence from the braid word, to decide order from the cutting sequence and to put a braid, 68 FENN, GREENE, ROLFSEN, ROURKE, AND WIEST given in terms of standard twist generators, into its left-consistent canonical form. Cutting sequences and curve diagrams. A cutting sequence is a finite word χ in the letters 0, . . . , n, 0, . . . , n + 1, ↑ and ↓ such that (i) χ starts with 0 and ends with n + 1, (ii) each of the letters 0, . . . , n + 1 occurs precisely once in χ, (iii) in the word χ numbers and arrows alternate, with the single possible exception that strings of the form i i + 1 or i + 1 i (i = 0, . . . , n) may occur. Consider now a curve diagram Γ. It consists of three types of subcurves: Curves in the upper half plane, curves in the lower half plane, and straight line segments in the real line. Note that curves in the upper or lower half plane may be replaced by semicircles since they are determined by their end points. For convenience we rescale the curve diagram so that it goes from 0 to n + 1 and the n holes are the integers 1, 2, . . . , n. Going along Γ we can read off a cutting sequence, by reading an ↑ or ↓ for every curve in the upper or lower half plane respectively, an i (i ∈ {0, . . . , n + 1}) for every intersection with the integer i in the real line (so underlined integers correspond to holes), and an i for every intersection with the real interval (i, i + 1). It is easy to check that a word obtained in this way is indeed a cutting sequence. For example the curve diagram representing σ1 in Figure 2 is coded as 0 ↑ 2 1 ↓ 3 4, whereas σ1 σ2−1 is coded 0 ↑ 1 ↓ 3 ↓ 1 ↓ 3 ↑ 2 ↑ 4. We define a reduction of a cutting sequence to be a replacement of the sequence by a shorter one, according to the one of the following rules (where l denotes ↑ or ↓, and i ∈ {0, . . . n}). • • • • i l i → i, i + 1 l i → i + 1, i l i → i, i l i + 1 → i + 1, ↓ i ↓ → ↓, ↑ i ↑ → ↑, i l i → i, i l i + 1 → i i + 1, i + 1 l i → i + 1 i A cutting sequence is called reduced if it allows no reduction. Proposition A.1. Every braid on n strings has a unique reduced cutting sequence. Proof. Let χ be a cutting sequence of a curve diagram Γ of the braid. We observe that a reduced version χ0 of χ is the same as the cutting sequence of a curve diagram Γ0 , where Γ0 is obtained by reducing Γ with respect to the trivial curve diagram E. From Proposition 2.3 we deduce that any two reduced cutting sequences χ0 and χ00 must come from curve diagrams which are equivalent with respect to E. Therefore χ0 and χ00 must agree. ORDERING THE BRAID GROUPS 69 The reduced curve diagram can be reconstructed from the reduced cutting sequence. Thus the cutting sequence classifies the curve diagram, and hence the braid. This is most easily seen by using pen and paper. One reads the cutting sequence, and for every number symbol one encounters, draws one arc in the diagram. If the cutting sequence is reduced, then this involves no choices. Below we shall give an algorithm to do this which is more suitable for computer implementation. Note that it is easy to construct reduced cutting sequences which do not come from curve diagrams. The pen and paper method can also be used to decide whether a cutting sequence does correspond to a curve diagram. Again we give a more formal algorithm below which will do this. Reading the cutting sequence from the braid word. We next show how to convert a braid defined in terms of the twist generators σi±1 into a reduced cutting sequence. We do this inductively by defining how σi and σi−1 act on reduced cutting sequences and then let the whole word act on the trivial sequence 0 1 . . . n n + 1. Algorithm A.2. Suppose a braid β has reduced cutting sequence χ. Then a cutting sequence of βσi is obtained by simultaneously making the following replacements everywhere in the word χ. These rules are to be interpreted as simultaneous, not sequential, replacements. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) i → i + 1, i + 1 → i, ↓ (i) → ↓ i − 1 ↑ (i + 1), (i) ↓ → (i + 1) ↑ i − 1 ↓, i − 1 (i) → i − 1 ↑ (i + 1), (i) i − 1 → (i + 1) ↑ i − 1, ↑ (i) → ↑ (i + 1), (i) ↑ → (i + 1) ↑, ↓ (i + 1) → ↓ (i), (i + 1) ↓ → (i) ↓, i + 2 (i + 1) → i + 2 ↓ (i), (i + 1) i + 2 → (i) ↓ i + 2, ↑ (i + 1) → ↑ i + 1 ↓ (i), (i + 1) ↑ → (i) ↓ i + 1 ↑, ↓ i ↑ → ↓ i − 1 ↑ i ↓ i + 1 ↑, ↑ i ↓ → ↑ i + 1 ↓ i ↑ i − 1 ↓. Note. In rules (ii)-(vii), rule (i) is being applied, and its application is indicated by brackets. Replacements of symbols other than i, i + 1 depend on context, eg rule (ii) says that if ↓ is followed by i, then it is to be replaced by ↓ i − 1 ↑, and the i is replaced by i + 1, by (i). So ↓ i turns into ↓ i − 1 ↑ i + 1. The rules for the action of σi−1 are obtained by interchanging the symbols ↑ and ↓ everywhere in this list (i.e. replacing up- by down-, and down- by up-arrows). The resulting cutting sequence can then be reduced, to obtain the reduced cutting sequence of the braid βσi or βσi−1 . We can now deduce an effective algorithm to decide whether a given braid is positive, trivial, or negative: 70 FENN, GREENE, ROLFSEN, ROURKE, AND WIEST Algorithm A.3. (To decide if a given braid is positive, trivial, or negative.) Use Algorithm A.2 to calculate the reduced cutting sequence of the braid. The braid is positive if and only if the first arrow in this sequence is an up-arrow ↑. Recovering the curve diagram from the cutting sequence. We now show how to recover a reduced curve diagram from its associated cutting sequence. At the same time this will provide an effective algorithm to decide if a given cutting sequence corresponds to a curve diagram. To make precise the problem here, we define the real cutting sequence of a curve diagram to be the cutting sequence, with the non-underlined integers replaced by real numbers specifying the precise intersection point of the curve diagram with the real line, up to order preserving bijections. (Taking the integer part of all numbers in the real cutting sequence we retrieve the cutting sequence.) Given the real cutting sequence, we can immediately construct the curve diagram. Moreover it is trivial to check if a real cutting sequence corresponds to an (embedded) curve diagram: One just checks that (1) if i i + 1 or i + 1 i occurs in the sequence then no real number in (i, i + 1) occurs, (2) the numbers on each side of two arrows of the same type correspond to nested intervals (so that the corresponding curves do not intersect). So we need an algorithm to reconstruct the real cutting sequence from the cutting sequence or equivalently to decide for each i the order in which the corresponding points actually occur in R. Algorithm A.4. Suppose the letter i (i ∈ {0, . . . , n}) appears in two different places, say in the rth and sth position, in the cutting sequence. To decide which one represents the smaller number in the interval (i, i + 1) in the real cutting sequence proceed as follows. Since the cutting sequence is reduced, there are two arrows in opposite direction adjacent to each of the letters i. Starting at the rth letter we read the sequence either forwards or backwards. We define the up-string at the rth place to be the word obtained from the cutting sequence by reading forwards or backwards, starting at the rth letter, up to the next underlined number, with the reading direction specified by the requirement that the the first two letters read should be i ↑. Similarly, we define the down-string at the rth place by reading in the opposite direction, such that the resulting word starts with i ↓, again up to the next underlined number. We compare the up-string at the rth with that at the sth place, and the down-string at the rth with that at the sth place. They cannot both agree, for if they did, the curve diagram would have two curves with the same endpoints. We now manipulate the up- and down strings as follows: Firstly, we increase all non-underlined integers by 21 . Then we remove the underline from all underlined integers. We obtain sequences of the form x0 l x1 l ORDERING THE BRAID GROUPS 71 . . . l xl−1 l xl , where l ∈ N, x0 = i + 21 , x1 , . . . , xl−1 ∈ { 21 , 1 12 , . . . , n + 21 }, and xl ∈ {0, . . . n + 1}. From this we can construct a sequence of numbers in {1, 1 12 , . . . , n − 1 2 , n}, called the cyclically associated sequence, as follows. For every string xj ↑ xj+1 we write down the unique representative in { 12 , 1, . . . , n, n + 21 } of xj+1 − xj + (n + 1)Z ∈ R/(n + 1)Z; for every string xj ↓ xj+1 we write down the unique representative in { 21 , 1, . . . , n, n + 12 } of xj − xj+1 + (n + 1)Z ∈ R/(n + 1)Z. Altogether, this yields a sequence of length l. We now define an up-string u to be cyclically lexicographically larger than another up-string u0 , if the cyclically associated sequence of u is lexicographically larger than the one of u0 .3 The geometric interpretation is that the curve diagram has two line segments starting in the real interval (i, i + 1), going into the upper half plane. The line segment representing the cyclically lexicographically larger up-string is the one turning ‘more to the left’. Since the two line segments must be disjoint (being part of the curve diagram), the starting point of the curve segment yielding the cyclically lexicographically larger up-string must represent a smaller real number in the real cutting sequence. Similarly, we define a cyclic lexicographic ordering on the downstrings; this time, the starting point of a curve segment which gives rise to a cyclically lexicographically larger down-string than another curve segment must represent a larger real number in the real cutting sequence. End of Algorithm A.4. To summarise, we have found an algorithm for reconstructing the real cutting sequence from the cutting sequence: Given any two places in the cutting sequence where the letter i occurs, we compare the up-strings at these places. If they agree, we compare the down-strings instead. In either case we can work out the cyclically associated sequences, and then decide which of the two letters i represents the smaller number in the interval (i, i + 1) in the real cutting sequence. An algorithm to determine order from the cutting sequence. Algorithm A.4 also allows us to decide which of two given reduced cutting sequences represents the larger braid. If the two sequences agree on some initial segment, then we remove the underlines from all underlined numbers (except the first letter 0) that lie in this segment. Then we reduce the resulting two sequences. We obtain two new sequences whose initial segments up to the first underlined numbers do not agree. If they differ already on the second letter (after 0), then we know which one is larger. Otherwise, we work out which of them is cyclically lexicographically larger, using Algorithm A.4. 3 Cyclic lexicographic order is used by Birman and Series [3]. 72 FENN, GREENE, ROLFSEN, ROURKE, AND WIEST The algorithm to determine left-consistent canonical form. We are finally ready to describe our algorithm to calculate the left-consistent canonical form of a braid. The input is a braid β represented as a word w in the twist generators σi±1 . The output is the same braid in left-consistent canonical form of β, again given as a word in the σi±1 . The algorithm proceeds by repeating the main step (described below) after each repetition we have a word W and a cyclically reduced cutting sequence χ which are both modified at the next repetition. Start. We start with W the trivial word, and χ the reduced cutting sequence of β calculated using Algorithm A.2. Finish. If the reduced cutting sequence χ is 0 1 . . . n n + 1, then the algorithm stops, and the inverse of the word W is the desired canonical word. Main step. If the reduced cutting sequence starts 0 1 . . . i ↑, with i < n+1, then we hunt for subwords of the following forms (i) (ii) (iii) (iv) i ↑ a1 ↓ a2 ↑ . . . l al−1 l al or i ↑ a1 ↓ a2 ↑ . . . l al−1 l al or al l al−1 l . . . ↑ a2 ↓ a1 ↑ i or al l al−1 l . . . ↑ a2 ↓ a1 ↑ i, where the a1 , . . . , al−1 are not equal to i and not underlined, and al 6= i, i+1. (If the reduced cutting sequence starts 0 . . . i ↓, then we hunt for subwords like i ↓ a1 ↑ . . . l al−1 l al instead.) We shall call these words useful subwords, because they correspond to useful arcs. We consider the set of all useful subwords, and we want to identify the ‘leftmost one’, i.e. the one whose letter i represents the leftmost point in the interval (i, i + 1). If one of them starts or ends with a letter i, i.e. if one of them is of type (ii) or (iv), then this is it. If not, then we can use Algorithm A.4 to determine the leftmost one. When we have found the leftmost useful subword, we modify it as follows. If it is of type (i) or (ii), then we write it backwards, so that it starts with the letter al . Irrespectively of the type of the useful subword, we remove the underline from the letter al . Then we let c := al , replace all letters ak (k ∈ {1, . . . , l}) with ak > c by ak − 1 (e.g. al turns into al − 1), and reduce the resulting sequence. By doing this, we obtain a modified sequence a00 l a01 l . . . l a0l0 , possibly with the letter a0l0 = i underlined. We now multiply W on the right by a word v1 . . . vl , where vk is determined by a0k−1 , a0k , and the arrow in between a0k−1 and a0k as follows: (i) If the modified leftmost useful subword contains the string a0k−1 ↑ a0k , and a0k−1 < a0k , then vk = σa0k−1 +1 . . . σa0k ; ORDERING THE BRAID GROUPS 73 (ii) If the modified leftmost useful subword contains the string a0k−1 ↑ a0k , and a0k−1 > a0k , then vk = σa−1 . . . σa−1 0 0 +1 ; k−1 k (iii) If the modified leftmost useful subword contains the string a0k−1 ↓ a0k , and a0k−1 < a0k , then vk = σa−1 . . . σa−1 0 0 ; +1 k−1 k (iv) If the modified leftmost useful subword contains the string a0k−1 ↓ a0k , and a0k−1 > a0k , then vk = σa0k−1 . . . σa0k +1 . The word v1 . . . vl represents the slide of a hole back along the leftmost useful arc. Finally, we calculate the new reduced cutting sequence after this slide. This can be done by letting the word v1 . . . vl act on the reduced cutting sequence, as described above. (An alternative method would be to remove the underline from the letter al , underline the unique letter i which belongs to the leftmost useful subword instead, carefully relabel the cutting sequence, using Algorithm A.4, and then reduce the resulting cutting sequence.) End of main step. The proof of Theorem 4.2 implies that the algorithm stops after a finite number of repetitions of the main step. References [1] G.M. Bergman, Right orderable groups that are not locally indicable, Pacific J. Math., 174 (1991), 243-248. [2] J. Birman, Braids, links, and mapping class groups, Annals of Math. Studies, 82, Princeton University Press, Princeton, 1975. [3] J.S. Birman and C. Series, An algorithm for simple curves on surfaces, J. London Math. Soc., 29(2) (1984), 331-342. [4] R.G. Burns and V.W.D. Hale, A note on group rings of certain torsion free groups, Canad. Math. Bull., 15 (1972), 441-445. [5] P. Dehornoy, Braid groups and left distributive operations, Trans. AMS, 345 (1994) 115-150. [6] [7] , From large cardinals to braids via distributive algebra, J. Knot Theory and its Ramifications, 4 (1995), 33-79. , A fast method of comparing braids, Adv. in Math., 125 (1997), 200-235. [8] E.A. Elrifai and H.R. Morton, Algorithms for positive braids, Quart. J. Math. Oxford, 45 (1994), 479-497. [9] D.B.A. Epstein et al, Word processing in groups, Jones & Bartlett, 1992. [10] E.A. Gorin and V.Ja. Lin, Algebraic equations with continuous coefficients, and certain questions of the algebraic theory of braids, Math USSR-Sb., 7 (1969), 569596. [11] R Laver, Braid group actions on left-distributive structures and well-orderings in the braid group, J. Pure Appl. Algebra, 108 (1996), 81-98. 74 FENN, GREENE, ROLFSEN, ROURKE, AND WIEST [12] L. Mosher, Mapping class groups are automatic, Math. Research Letters, 1 (1994), 249-255. [13] , Mapping class groups are automatic, Annals of Math., 142 (1995), 303-384. [14] , A user’s guide to the mapping class group: Once punctured surfaces, MSRI preprint. [15] D. Rolfsen and Jun Zhu, Braids, orderings and zero divisors, submitted to J. Knot Theory and its Ramifications. [16] C. Rourke and B. Wiest, Order automatic mapping class groups, to appear; http://www.maths.warwick.ac.uk/˜cpr/ftp/ordaut.ps [17] B. Wiest, Dehornoy’s ordering of the braid groups extends the subword ordering, Pacific J. Math., 191(1) (1999), 183-188. Received February 17, 1998. University of Sussex Falmer, Brighton BN1 9QH UK E-mail address: [email protected] Radan Computational, Ensleigh House Granville Road, Bath BA1 9BE UK E-mail address: [email protected] University of British Columbia Vancouver, B.C. V6T 1Z2 Canada E-mail address: [email protected] University of Warwick Coventry CV4 7AL UK E-mail address: [email protected] CMI, Université de Provence 13453 Marseille cedex 13 France E-mail address: [email protected] PACIFIC JOURNAL OF MATHEMATICS Vol. 191, No. 1, 1999 NUMERICAL SEMIGROUPS GENERATED BY INTERVALS P.A. Garcı́a-Sánchez and J.C. Rosales We study numerical semigroups generated by intervals and solve the following problems related to such semigroups: the membership problem, give an explicit formula for the Frobenius number, decide whether the semigroup is a complete intersection and/or symmetric, and computation of the cardinality of a (any) minimal presentation of this kind of numerical semigroups. A numerical semigroup is a finitely generated subsemigroup of the set of nonnegative integers N, such that the group generated by it is the set of all integers Z. In this paper we study the semigroups generated by intervals of nonnegative integers, that is to say, semigroups of the form S = ha, a + P 1, . . . , a + xi = { xi=0 ni (a + i) : ni ∈ N} ⊆ N. Note that if x ≥ a, then S = {a, a + 1, . . . } = a + N = ha, a + 1, . . . , 2a − 1i; thus we may assume that x ≤ a − 1. For a semigroup of this kind we solve the following problems: 1) Membership problem. An element n ∈ N belongs to S = ha, a + 1, . . . , a + xi if and only if n mod a ≤ b na cx, where b na c is quotient of the integer division of n by a, and n mod a denotes the remainder of this division, n − b na ca. 2) Computation of the Frobenius number of the semigroup. The Frobenius number of a numerical semigroup (also known as the conductor of the semigroup) is the greatest integer not belonging to the given semigroup. The Frobenius number of S = ha, a+1, . . . , a+xi is d a−1 x ea−1, where dqe denotes the least integer greater than or equal to q ∈ Q+ . 3) Symmetry of the semigroup. A numerical semigroup T with Frobenius number C is symmetric if and only if for each z ∈ Z we have that either z ∈ S or C − z ∈ S. These kinds of semigroups are specially interesting in Ring Theory as Kunz shows in [7]. The semigroup S = ha, a + 1, . . . , a + xi is symmetric if and only if a ≡ 2 mod x (here a ≡ b mod c denotes the fact a − b = kc for some integer k). 4) Cardinality of a minimal presentation of S. The semigroup S = ha, a+ 1, . . . , a+xi is isomorphic to Nx+1 /σ, where σ is the kernel congruence of the semigroup morphism ϕ:N x+1 → S, ϕ(n0 , . . . , nx ) = x X i=0 75 ni (a + i), 76 P.A. GARCÍA-SÁNCHEZ AND J.C. ROSALES that is to say a σ b if and only if ϕ(a) = ϕ(b). A minimal presentation of S is a minimal system of generators of the congruence σ. In this paper we show that the cardinality of a minimal presentation of S is x(x − 1) + x − ((a − 1) mod x). 2 5) Complete intersection semigroups. A numerical semigroup is a complete intersection if the cardinality of a minimal presentation plus one equals the cardinality of a minimal system of generators of the given semigroup (see [6]). We show that the semigroup S = ha, a + 1, . . . , a + xi is a complete intersection if and only if S is ha, a + 1i or h2k, 2k + 1, 2k + 2i. The point of departure to solve these problems is the following lemma. Lemma 1. Let S = ha, a + 1, . . . , a + xi be a numerical semigroup with 1 ≤ x < a. Then, n ∈ S if and only if n = qa + i with q ∈ N and i ∈ {0, . . . , qx}. P Proof. If n ∈ S then there exist n0 , . . . , nx ∈ N such that n = xj=0 nj (a+j). P P P P Thus, n = ( xj=0 nj )a + xj=1 nj j. Take q = xj=0 nj and i = xj=1 nj j ≤ Px j=0 nj x = qx. Now, assume that n = qa + i, with 0 ≤ i ≤ qx. We distinguish two possible cases: 1) If i = qx then n = q(a + x) ∈ S. 2) If i = kx + r, with 0 ≤ k < q and 0 ≤ r ≤ x − 1, then n = qa + kx + r = (q − k − 1)a + k(a + x) + a + r ∈ S. From this membership characterization, we can derive the following characterization which is easier to check and it is what we will use later in the paper. Corollary 2. Let S = ha, a + 1, . . . , a + xi be a numerical semigroup with 1 ≤ x < a. Then, n ∈ S if and only if (n mod a) ≤ b na cx. Proof. If n ∈ S, then using the previous lemma, there exists q ∈ N and 0 ≤ i ≤ qx such that n = qa + i. Besides, n = b na ca + (n mod a). Thus, n mod a = qa + i − b na ca = (q − b na c)a + i, and since q ≤ b na c, we get that n mod a ≤ i ≤ qx ≤ b na cx. Now, assume that (n mod a) ≤ b na cx. Since n = b na ca + (n mod a) and 0 ≤ n mod a ≤ b na cx, applying the previous result, we get that n ∈ S. For every n ∈ S, the Apéry set (see [1]) associated to n is defined as S(n) = {s ∈ S : s − n 6∈ S}. Using last result, we can characterize the elements belonging to S(a). This characterization is going to play an important role in the rest of the paper. NUMERICAL SEMIGROUPS GENERATED BY INTERVALS 77 Corollary 3. Let S = ha, a + 1, . . . , a + xi be a numerical semigroup with 1 ≤ x < a and let n ≥ a. Then, n ∈ S(a) if and only if (n mod a) ∈ {(b na c − 1)x + 1, . . . , b na cx}. Proof. The element n ∈ S(a) if and only if n ∈ S and n − a 6∈ S. By the previous result this occurs if and only if (n mod a) ≤ b na cx and ((n − n−a n a) mod a) > b n−a a cx. But (n − a) mod a = n mod a and b a c = b a c − 1 n (note that n ≥ a). Hence, n ∈ S(a) if and only if n mod a ∈ {(b a c − 1)x + 1, . . . , b na cx}. We can explicitly construct S(a) as the next corollary shows. Corollary 4. Let S = ha, a + 1, . . . , a + xi be a numerical semigroup with 1 ≤ x < a. Then, S(a) = {qa + (q − 1)x + r : 1 ≤ r ≤ x, q ∈ N and 0 ≤ (q − 1)x + r < a}. Proof. Take n = qa + (q − 1)x + r such that 1 ≤ r ≤ x, q ∈ N and 0 ≤ (q−1)x+r < a. Then, b na c = q and n mod a = (q−1)x+r = (b na c−1)x+r ∈ {(b na c − 1)x + 1, . . . , b na cx}. Using the previous result, we get that n ∈ S(a). Now, take n = b na ca + (n mod a) ∈ S(a). Then, by the previous corollary, n mod a ∈ {(b na c−1)x+1, . . . , b na cx} and therefore n mod a = (b na c−1)x+r, with r ∈ {1, . . . , x}. Taking q = b na c we are done. If S is a numerical semigroup then the set N \ S is finite, because the group spanned by S is Z. As we have mentioned before, the maximum of this set is called the Frobenius number of the semigroup, which we denote by C(S). It is well known (see [2]) that C(S) = max(S(a)) − a. Thus, if we want to compute C(S), we have to determine the greatest element in S(a). This is performed in the next result. Corollary 5. Let S = ha, a + 1, . . . , a + xi be a numerical semigroup with 1 ≤ x < a. Then, C(S) = d a−1 x ea − 1. Proof. We must determine the maximum of S(a) = {qa + (q − 1)x + r : 1 ≤ r ≤ x, q ∈ N and 0 ≤ (q − 1)x + r < a}. The maximum is reached when (q − 1)x + r = a − 1. The element a − 1 is equal to b a−1 x cx + ((a − 1) mod x). Two possibilities arise: • If (a − 1) mod x 6= 0 then take (q − 1) = b a−1 x c and r = (a − 1) mod x. The greatest element in S(a) is qa + (q − 1)x + r = (b a−1 x c + 1)a + a − 1. • If (a − 1) mod x = 0 then, since r must be in {1, . . . , x}, write a − 1 as a−1 a−1 = (b a−1 x c−1)x+x. Take q−1 = b x c−1 and r = x. In this case, the greatest element in S(a) is qa + (q − 1)x + r = qa + (q − 1)x + x = b a−1 x ca + a − 1. Both cases are represented by d a−1 x ea+a−1, which is the greatest element in S(a). 78 P.A. GARCÍA-SÁNCHEZ AND J.C. ROSALES Note that this implies that for the numerical semigroups generated by intervals there exists an explicit formula to compute the Frobenius number of the semigroup. It seems that there is no known formula for the general case. Nevertheless, for some specific cases there exists an explicit formula. For instance, in [4], a formula for the Frobenius number of numerical semigroups generated by up to three elements and of symmetric numerical semigroups generated by up to four elements is given (see also [5] for more references). Another characterization of symmetric numerical semigroups is the following (see [3]). The numerical semigroup S is symmetric if the greatest element, w, of S(a) satisfies the condition that for every s ∈ S(a), the element w − s is in S. We use this result to give a characterization of the numerical semigroups generated by intervals that are symmetric. Theorem 6. Let S = ha, a + 1, . . . , a + xi = 6 N be a numerical semigroup with 1 ≤ x < a. Then, S is symmetric if and only if a ≡ 2 mod x. Proof. We can assume that x ≥ 2, since if x = 1 then S is generated by two relatively prime elements and in this case it is well known that S is symmetric (see for instance [5]). Let w be the greatest element in S(a). We already know that w = qa + (q − 1)x + r with (q − 1)x + r = a − 1. If S is symmetric, since 1 6∈ S, then qa + (q − 1)x + r − 1 cannot be in S(a). Hence, qa + (q − 1)x + r − 1 ∈ S \ S(a), which means that n = (q − 1)a + (q − 1)x + r − 1 = qa + (q − 1)x + r − 1 − a ∈ S. Since n mod a = (q − 1)x + r − 1, b na c = q−1 and n mod a must be less than or equal to b na cx, we get that r−1 must be zero. Hence, a − 1 = (q − 1)x + 1, which means that a ≡ 2 mod x. If a ≡ 2 mod x, then (a − 1) ≡ 1 mod x, and therefore (a − 1) mod x = 1, which from the first case in the proof of the last corollary implies that r = 1. Consequently, w = qa + (q − 1)x + 1. Take 0 6= m = ka + (k − 1)x + i ∈ S(a), that is to say 0 6= k ∈ N, 0 ≤ (k − 1)x + i < a and i ∈ {1, . . . , x}. Then, w − m = (q − k)a + (q − k − 1)x + (x − i + 1). Note that x − i + 1 ∈ {1, . . . , x}, q − k ∈ N and 0 ≤ (q − k − 1)x + (x − i + 1) = (q − k)x + 1 − i < a. Hence, w − m ∈ S(a) ⊂ S for all m ∈ S(a) and this means that S is symmetric. The rest of the paper is devoted to computing the cardinality of a minimal presentation of the numerical semigroup S = ha, a + 1, . . . , a + xi with 1 ≤ x < a. It can be shown (see [6] for example) that for a numerical semigroup the cardinality of a set of generators of σ, the kernel congruence of ϕ, is greater than or equal to the number of generators of the semigroup minus one (in our case, this amount is x). When this lower bound is reached, the semigroup is a complete intersection. These semigroups are always symmetric. In [8], the first author gives an algorithm to compute a system of generators, ρ, for σ with minimal cardinality. From the results given in that paper, it is determined that the concepts of system of generators for σ with NUMERICAL SEMIGROUPS GENERATED BY INTERVALS 79 minimal cardinality and minimal system (with respect to the inclusion) of generators of σ coincide. Next, we give a sketch of this construction, which is needed to count the elements in a minimal presentation of S. For every n ∈ S, we define the graph Gn = (Vn , En ), as Vn = {a + i ∈ {a, . . . , a + x} : n − (a + i) ∈ S}, En = {[a + i, a + j] : n − ((a + i) + (a + j)) ∈ S, i 6= j ∈ {0, . . . , x}}. We define ρn as: 1) If Gn is not connected and G1n = (Vn1 , E1n ), . . . , Grn = (Vnr , Ern ) are its connected components, then for every 1 ≤ i ≤ r we select an element αi = (n0i , . . . , nxi ) ∈ Nx+1 \ {0} such that ϕ(αi ) = n and nki = 0 for all a + ki 6∈ Vni . Define ρn = {(α1 , α2 ), (α1 , α3 ), . . . , (α1 , αr )}. 2) If Gn is connected, we define ρn = ∅. S The set ρ = n∈S ρn is a system of generators for σ with minimal cardinality (see [8]). Thus, we must look for the elements n ∈ S such that Gn is not connected. Example 7. Let S = h6, 7, 8i. The vertices of G14 are {6, 7, 8} and the only edge of G14 is [6, 8]. Thus G14 has two connected components: One is the vertex 7 and the other is the edge [6, 8]. The elements (0, 2, 0) and (1, 0, 1) are in N3 and verify that ϕ(0, 2, 0) = ϕ(1, 0, 1) = 14 (observe that the first and last coordinates of (0, 2, 0) are zero and that the second coordinate of (1, 0, 1) is zero). Thus ρ14 = {((0, 2, 0), (1, 0, 1))}, meaning that 2 × 7 = 6 + 8 is a relation on S. Theorem 8. Let S = ha, a + 1, . . . , a + xi be a numerical semigroup with 1 ≤ x < a and ρ as before. Then, x(x − 1) + x − ((a − 1) mod x). 2 Proof. From the proof of Lemma 1, it is derived that if n ∈ S then n can be expressed in one of the following ways: • n = ka + l(a + x) for some nonnegative integers k, l. • n = ka + l(a + x) + (a + i) for some nonnegative integers k, l and i ∈ {1, . . . , x − 1}. Hence, if Gn is not connected, then the set {a, a + x} ∩ Vn is not empty. In the construction of ρ we take G1n to be the connected component containing a, if a ∈ Vn . If a is not a vertex of Gn then we take G1n to be the connected component containing a + x (which must be in Vn ). We are going to count the elements in ρ which come from the fact that a + i, 1 ≤ i < x is neither in the connected component containing a nor in the connected component containing a + x (if a or a + x is not in Vn this is translated to the fact that a + i is not in Vn1 ). Next, we will count the elements in ρ which arise when #ρ = 80 P.A. GARCÍA-SÁNCHEZ AND J.C. ROSALES a and a + x are in different connected components (that is to say, a + x not in Vn1 ). With this, we count all the elements belonging to ρ. • First, let us assume that Gn is not connected and there is a + i ∈ Vn such that a + i is not in the connected component(s) containing {a, a + x} ∩ Vn . Hence, n − (a + i + a) 6∈ S and n − (a + i + a + x) 6∈ S, which implies that n = w + (a + i) with 0 6= w ∈ S(a) ∩ S(a + x). It is easy to check, from the description of S(a) given in Corollary 4, that S(a) ∩ S(a + x) = {0, a + 1, . . . , a + x − 1}. Thus, there exists j ∈ {1, . . . , x − 1} such that n = (a + j) + (a + i). Note that the reverse is also true: If n = (a + i) + (a + j) with i, j ∈ {1, . . . , x − 1}, then the elements a and a + i are not connected in Gn and the same holds for a + x and a + i. This means that every expression of the form n = (a + i) + (a + j) with 1 ≤ i, j ≤ x − 1 yields a new element in ρn (the element (α1 , αt ), where Gtn is the connected component of Gn containing a + i). This implies that from these graphs we get as many elements in ρ as pairs (i, j) with 1 ≤ i, j ≤ x − 1. This amount is x(x − 1)/2. • Now, let us count the elements in ρ coming from non-connected graphs Gn such that a and a + x are in different connected components. Since a and a + x are in different connected components of Gn , the element n can be expressed as n = w + (a + x), where w ∈ S(a). Note also that n 6∈ S(a). Thus, we must find the elements w in S(a) such that w + (a + x) − a = w + x ∈ S. By the proof of Corollary 5, depending on r = (a − 1) mod x, the maximum element in S(a) is qa + (q − 1)x + r, if r 6= 0, or qa + (q − 1)x + x, if r = 0, where q = d a−1 x e. 1) If r 6= 0, then q = b a−1 c + 1 ≥ 2. Let us show that the element x w = ka + (k − 1)x + i ∈ S(a) (1 ≤ i ≤ x and 0 ≤ (k − 1)x + i < a) satisfies that w + x is not in S when k ≤ q − 2. Note that (k − 1)x + i + x ≤ (k + 1)x ≤ (q − 1)x = b a−1 x cx ≤ a − 1 < a. Thus, (w + x) mod a = kx + i and b w+x cx = kx. Since i ≥ 1, by a Corollary 2, w + x 6∈ S. Let us show that w = (q − 1)a + (q − 2)x + i ∈ S(a) verifies that w + x 6∈ S when i ≤ r. Note that (q − 2)x + i + x = (q − 1)x + i ≤ (q − 1)x + r = a − 1. Thus, (w + x) mod a = (q − 1)x + i and b w+x a cx = (q − 1)x, which by Corollary 2 implies that w + x 6∈ S. In addition, let us prove that if w ∈ {qa + (q − 1)x + 1, . . . , qa + (q − 1)x + r}, then a and a + x are in the same connected component of Gw+(a+x) . Take w = qa+(q −1)x+i in this set. Then, w +(a+x) = q(a + x) + (a + i), which means that [a + x, a + i] ∈ E(Gn ). Besides, w + (a+ x) = (q + 1)a+qx + i = (q +1)a+((q − 1)x+ r)+ x− r + i = (q + 1)a + (a − 1) + x − r + i = qa + (a + i) + (a + (x − r − 1)) (observe NUMERICAL SEMIGROUPS GENERATED BY INTERVALS 81 that 0 ≤ x − r − 1 ≤ x). This implies that [a, a + i] ∈ E(Gn ) and consequently a and a + x are connected in Gn . Hence, the elements w we are interested in must be in the set R = {(q − 1)a + (q − 2)x + r + 1, . . . , (q − 1)a + (q − 2)x + x}. Let us show that, as a matter of fact, for the elements n ∈ S of the form n = w + (a + x), with w in the previous set, the graph Gw+(a+x) has no path connecting a and a + x. In order to show this, it is enough to prove that if w = (q − 1)a + (q − 2)x + r + i ∈ R, n − ((a + k) + (a + j)) ∈ S and j ≥ i then k must be greater or equal than i. The element m = n − (a + k + a + j) is equal to (q − 2)a + (q − 1)x + r + i − k − j. We distinguish two cases depending on the value of q − 1: – If q − 1 ≥ 2, then (q − 1)x + r + i ≥ 2x ≥ k + j and therefore (q − 1)x + r + i − k − j ≥ 0. Besides, (q − 1)x + r + i − k − j = (a − 1) + i − k − j ≤ a − 1 since i − j ≤ 0. Hence, m mod a = (q − 1)x + r + i − k − j and b m a cx = (q − 2)x. Using Corollary 2, we get that since m = n − (a + j + a + k) ∈ S, (q−1)x+r+i−j−k ≤ (q−2)x, that is to say, x+r+i−k−j ≤ 0, and this occurs if and only if (x − j) + r + i ≤ k which implies that k ≥ i, since x − j + r ≥ 0. – If q − 1 = 1 then m = x + r + i − k − j ≥ 0, since m ∈ S and as before x + r + i − k − j ≤ a − 1. In this case, m mod a = x + r + i − k − j and b m a c = 0. Thus, m ∈ S implies that x + r + i − k − j ≤ 0, and this leads to k ≥ i. Taking all this into account we have as many new elements in ρ as elements has the set R, and this amount is x − r = x − ((a − 1) mod x). 2) If r = 0, then q = b a−1 x c ≥ 1. Let us show that the element w = ka + (k − 1)x + i ∈ S(a) (1 ≤ i ≤ x and 0 ≤ (k − 1)x + i < a) satisfies that w + x is not in S when k ≤ q − 1. This is due to the fact that (k − 1)x + i + x = kx + i ≤ (q − 1)x + i = a − 1 − x + i < a (recall that i ≤ x), and therefore (w + x) mod a = kx + i and b w+x a cx = kx, which, once more, implies that w + x 6∈ S. In this case, we get that w must be in the set R = {qa + (q − 1)x + 1, . . . , qa + (q − 1)x + x}. In the same way we did for the case r 6= 0, it can be shown that all these elements produce graphs such that a and a + x are in different connected components (the proof is the same but for the cases to distinguish, which are q − 1 ≥ 1 and q − 1 = 0). Again, we get x = x − 0 = x − ((a − 1) mod x) new elements in ρ. 82 P.A. GARCÍA-SÁNCHEZ AND J.C. ROSALES Counting all the elements in ρ, we have that #ρ = x(x − 1) + x − ((a − 1) mod x). 2 With this theorem it is easy to prove the next result. Corollary 9. Let S = ha, a + 1, . . . , a + xi be a numerical semigroup with 1 ≤ x < a and ρ as before. Then, S is a complete intersection if and only if one of the following cases occur: 1) S = ha, a + 1i (x = 1). 2) S = h2k, 2k + 1, 2k + 2i (x = 2 and a mod 2 = 0). Proof. The semigroup S is a complete intersection if and only if #ρ = x. Using previous theorem this occurs if and only if x(x − 1)/2 + x − ((a − 1) mod x) = x, and this happens if and only if x(x − 1)/2 = (a − 1) mod x. Thus, if S is a complete intersection then x(x − 1)/2 < x and this implies that x < 3, since x > 0. Hence, two cases may occur: 1) x = 1. In this case, S = ha, a + 1i. Note that if S is of this form then S is a complete intersection. 2) x = 2. Under this setting then S = ha, a+1, a+2i. Besides, if we want S to be a complete intersection then 2(2−1)/2+2−((a−1) mod 2) = 2 and this leads to a mod 2 = 0. Finally, we would like to thank the referee for his/her wise comments and suggestions. References [1] R. Apéry, Sur les branches superlinéaires des courbes algébriques, C. R. Acad. Sci. Paris, 222 (1946). [2] A. Brauer and J. E. Shockley, On a problem of Frobenius, J. Reine Angew. Math., 211 (1962), 215-220. [3] H. Bresinski, Symmetric semigroups generated by 4 elements, Manuscripta Math., 17 (1995), 81-95. [4] R. Fröberg, The Frobenius number of some semigroups, Communications in Algebra, 22(14) (1994), 6021-6024. [5] R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83. [6] J. Herzog, Generators and relations of abelian semigroups and semigroup rings, Manuscripta Math., 3 (1970), 175-193. [7] E. Kunz, The value-semigroup of a one-dimensional Gorenstein ring, Proc. Amer. Math. Soc., 25 (1973), 748-751. NUMERICAL SEMIGROUPS GENERATED BY INTERVALS 83 [8] J.C. Rosales, An algorithmic method to compute a minimal relation for any numerical semigroup, Inter. J. Algebra and Computation, 6(4) (1996), 441-455. Received January 6, 1998 and revised October 1, 1998. This paper was supported by the project DGES PB96-1424. Universidad de Granada Departamento de Álgebra E-18071 Granada Spain E-mail address: [email protected] Universidad de Granada Departamento de Álgebra E-18071 Granada Spain E-mail address: [email protected] PACIFIC JOURNAL OF MATHEMATICS Vol. 191, No. 1, 1999 A SELBERG INTEGRAL FORMULA AND APPLICATIONS Loukas Grafakos and Carlo Morpurgo We obtain a 3-fold Selberg integral formula. As a consequence we are able to compute the explicit value of the sharp constant in a trilinear fractional integral inequality due to Beckner. 1. Introduction. Multilinear fractional integral inequalities have been used in connection with restriction theorems of the Fourier transform and also in obtaining estimates for the k-plane and the x-ray transform. See for instance [C1], [C2], and [D]. In this article we are interested in a sharp form of a multilinear fractional integral inequality obtained by [B] (Theorem 6). P > 1, and 0 ≤ γij = Theorem ([B]). Let 1 < p1 , . . . , pk < ∞, kj=1 p−1 j γji < n be real numbers satisfying (0) X 1≤j≤k, j6=s γjs = 2n p0s 1 n and X γij + 1≤i<j≤k k X 1 = k, pj j=1 where pj and p0j are dual exponents. Then Z k k Y Y Y −γ ij (1) nk dx1 . . . dxk ≤ A(γij , n) fj (xj ) |xi − xj | kf kpj . R j=1 j=1 1≤i<j≤k Moreover, the best constant A(γij , n) in (1) is attained for the extremal functions fj (x) = C(1 + |x|2 )−n/pj up to a conformal automorphism. The second condition in (0) is necessary to ensure conformal invariance of the variational inequality (1). It is worth mentioning that the one dimensional form of inequality (1) above when all the exponents are equal was obtained by [C1] without sharp constants (and without the first restriction in (0)). The value of the best constant in (1) was computed in [B]: 1 P γ Z −k+ n Y ij n 1≤i<j≤k (2) A(γij , n) = |S | |ξi − ξj |−γij dξ1 . . . dξk , (S n )k 1≤i<j≤k 85 86 LOUKAS GRAFAKOS AND CARLO MORPURGO where |S n | = (4π)n/2 Γ(n/2)Γ(n)−1 is the Lebesgue measure of the unit sphere in Rn+1 . This formula brings a connection between multilinear fractional integral inequalities and Selberg integrals. Multiple integrals such as the one in (2) are known as Selberg’s integrals and their exact values are useful in representation theory and in mathematical physics. These integrals have only been computed in special cases, for instance by Selberg himself when n = 1 and γij = γ (see [Se]), or when n = 2 and γij = 1 (see [Ca]), but not in general. For a treatment of Selberg integrals, the reader could consult [Me], Section 17.11. The question we would like to address is the following: Question. Can the constant A(γij , n) be computed explicitly? In this paper we give an answer to this question when k = 3. We are able to compute the three-fold Selberg integral (2) when γ12 + γ23 + γ31 = n for n ≥ 1. Before we state our first result we would like to discuss the case k = 2. The bilinear version of (1) is the well known Hardy-Littlewood-Sobolev inequality Z (3) |x − y|−γ f1 (x)f2 (y)dxdy ≤ E(γ, p1 , p2 , n)kf1 kp1 kf2 kp2 R2n which holds when 1/p1 + 1/p2 > 1, 1/p1 + 1/p2 + d/n = 2, and 0 < γ < n. The sharp constant in inequality (3) was derived by [L] when p1 = p2 = 2n/(2n − γ) and also when p1 = 2 or p2 = 2. When p1 = p2 = 2n/(2n − γ), the sharp constant in (3) is Z |ξ − η|−γ dξdη, E(γ, p1 , p2 , n) = |S n |(γ−2n)/n (S n )2 which can be easily computed since Z n−γ −γ n−γ n/2 Γ 2 (4) |ξ − η| dξ = 2 π , n Γ n − γ2 S for all given η ∈ S n . We now turn our attention to the case k = 3. It turns out that in this case we can find a closed form for the constant in (2) when 1/p1 +1/p2 +1/p3 = 2 or, equivalently, when γ12 + γ23 + γ31 = n. It will be convenient to slightly change our notation in this case. We set −γ12 = d1 − n, −γ23 = d2 − n, and −γ31 = d3 − n. Now for 0 < d1 , d2 , d3 < n we denote by (5) Qd1 ,d2 ,d3 [f1 , f2 , f3 ] Z = |x − y|d1 −n |y − z|d2 −n |z − x|d3 −n f1 (x)f2 (y)f3 (z)dxdydz R3n A SELBERG INTEGRAL FORMULA AND APPLICATIONS 87 the trilinear fractional integral that appears in (1). With this notation, inequality (1) is just (6) Qd1 ,d2 ,d3 [f1 , f2 , f3 ] ≤ C(d1 , d2 , d3 , n)kf1 kp1 kf2 kp2 kf3 kp3 , where 0 < d1 , d2 , d3 < n are real numbers satisfying d1 + d2 + d3 > n, and p1 = 2n/(d1 + d3 ), p2 = 2n/(d1 + d2 ), p3 = 2n/(d2 + d3 ). The best constant in the inequality above can be written as (7) Z C(d1 , d2 , d3 , n) = |S n |− d1 +d2 +d3 n |ξ −η|d1 −n |η−ζ|d2 −n |ζ −ξ|d3 −n dξdηdζ. (S n )3 We now state our first result: Theorem 1. Let 0 < d1 , d2 , d3 < n, and d1 + d2 + d3 = 2n. Then, for any distinct x, y, z ∈ Rn , the following formula holds Z (8) |x − t|−d2 |y − t|−d3 |z − t|−d1 dt Rn = B(d1 , d2 , d3 , n)|x − y|d1 −n |y − z|d2 −n |z − x|d3 −n , where B(d1 , d2 , d3 , n) = π n/2 n−dj 2 . d Γ 2j 3 Y Γ j=1 Sn Similarly, for any distinct ξ, η, ζ ∈ we have Z (9) |ξ − τ |−d2 |η − τ |−d3 |ζ − τ |−d1 dτ Sn = B(d1 , d2 , d3 , n)|ξ − η|d1 −n |η − ζ|d2 −n |ζ − ξ|d3 −n . Corollary 1. Let 0 < d1 , d2 , d3 < n and d1 + d2 + d3 = 2n. Then the following Selberg integral formula holds: Z (10) |ξ − η|d1 −n |η − ζ|d2 −n |ζ − ξ|d3 −n dξdηdζ (S n )3 n n = |S |(2π) 3 Y Γ j=1 Γ n dj 2 , d − 2j and thus the exact value of the best constant in (6) when d1 + d2 + d3 = 2n is d 3 Y Γ 2j n n −1 (11) C(d1 , d2 , d3 , n) = (2π) |S | . dj Γ n − j=1 2 88 LOUKAS GRAFAKOS AND CARLO MORPURGO We point out that the kernel formula (8) is a trilinear version of the standard beta integral on Rn : Z (12) |x − t|−α1 |y − t|−α2 dt n R n−α2 α1 +α2 −n n−α1 Γ n/2 Γ 2 Γ 2 2 |x − y|n−α1 −α2 , =π α1 α2 2 Γ 2 Γ 2 Γ n − α1 +α 2 which is valid when 0 < α1 , α2 < n, α1 + α2 > n. It is still unclear to us whether or not there is a corresponding k-fold analogue of (8) and (9). 2. The proof of Theorem 1. Clearly both sides of (8) are invariant under translations, dilations, and rotations of x, y, z. Therefore, by a translation we can assume that z = 0, by a dilation that |y| = 1, and by a rotation that y = e1 = (1, 0, . . . , 0). Let R us denote by fb(ξ) = Rn e−2πiξ·x f (x)dx the Fourier transform of f . Recall that d d−n ∧ n/2−d Γ 2 |ξ|−d := c(d)|ξ|−d (|x| ) (ξ) = π n−d Γ 2 in the sense of distributions (see [GS]). After these reductions, we prove (8) by showing that the Fourier transform of both sides coincide. The function x → |x − e1 |d1 −n |x|d3 −n has Fourier transform ∧ (13) |x|d3 −n |x − e1 |d1 −n (ξ) = c(d3 )c(d1 )|ξ|−d3 ∗ |ξ|−d1 e−2πiξ·e1 Z = c(d3 )c(d1 ) |ξ − η|−d3 |η|−d1 e−2πiη·e1 dη n R Z n−d1 −d3 = c(d3 )c(d1 )|ξ| |ξ 0 − t|−d3 |t|−d1 e−2πi|ξ|t·e1 dt, Rn ξ0 where = ξ/|ξ|. Now, for given ξ find a rotation Aξ so that Aξ e1 = ξ 0 . −1 −1 0 −1 −2 0 Clearly |ξ 0 −t| = |e1 −A−1 ξ t|, |t| = |Aξ t|, and t·e1 = t·Aξ ξ = Aξ t·Aξ ξ . Hence, with s = A−1 ξ t the expression in (13) is equal to Z −2 n−d1 −d3 c(d3 )c(d1 )|ξ| |e1 − s|−d3 |s|−d1 e−2πis·Aξ ξ ds Rn n−d1 −d3 b = c(d3 )c(d1 )|ξ| h(A−2 ξ ξ), where h(t) := |t − e1 |−d3 |t|−d1 . On the other hand, let us denote by g(x) the left hand side of (8) when z = 0 and y = (1, 0, . . . , 0). We have that ∧ gb(ξ) = h ∗ |t|−d2 (ξ) = c(n − d2 )b h(ξ)|ξ|d2 −n . A SELBERG INTEGRAL FORMULA AND APPLICATIONS 89 Using that d1 + d2 + d3 = 2n and that c(n − d)−1 = c(d) we deduce that the Fourier transforms of the two sides of (8) are equal if and only if (14) b h(ξ) = b h(A−2 ξ ξ) for almost all ξ ∈ Rn . We now use the fact that if a function is reflection invariant with respect to a hyperplane then so is its Fourier transform. Modulo rotations and translations it is enough to check this for hyperplanes of the form xj = 0. But the function h is constant along circles orthogonal to e1 ; in particular h is reflection invariant with respect to the hyperplanes xj = 0, for j = 2, 3, . . . , n, and hence so is b h. But A−2 ξ ξ can be obtained from ξ by finitely many reflections with respect to the above hyperplanes, and this concludes the proof of (8). To prove (9) we use the stereographic projection π̄ : Rn → S n . Recall that the Jacobian of π̄ is (15) |Jπ̄ (t)| = 2n (1 + |t|2 )−n , and that for any a, b in Rn we have (16) 1 1 |π̄(a) − π̄(b)| = 2|a − b|(1 + |a|2 )− 2 (1 + |b|2 )− 2 . Now let ξ = π̄(x), η = π̄(y), ζ = π̄(z), and τ = π̄(t) in the integral on the left hand side of (7). Using formulas (12) and (13) one can obtain (7) by simply rewriting (6) in terms of the coordinates ξ, η, ζ, τ . Proof of Corollary 1. Formula (10) follows by integrating (9) with respect to ξ, ζ, η and using (3). Formula (11) is immediate from (7). From analyticity considerations it follows that the upper bound for the dj in Corollary 1 can be extended to 2n instead of n. Of course this is not the case in Theorem 1 since the integral in (8) may diverge if dj ≥ n. 3. Application to a sharp Sobolev imbedding. The purpose if this section is to bring out some connections between multilinear integrals of type (1) with equal exponents and Sobolev imbeddings. These connections enter also in the context of conformal deformations of the metric structure of S n and spectral theory. Indeed, as shown in [Mo], multilinear fractional integrals arise as explicit computations of zeta functions of natural pseudodifferential operators, in the conformal class of the standard metric on S n . Let us discuss again the case k = 2. Inequality (3) can also be written as (17) kIα (f )kq ≤ Np,α,n kf kp , 90 LOUKAS GRAFAKOS AND CARLO MORPURGO where α −n = −γ, 1/q = 1/p−α/n, 1 < p, q < ∞, and Iα denotes fractional integration given by Z Γ n−α 2 |x − y|α−n f (y)dy. (18) Iα (f )(x) = α n/2 α 2 π Γ 2 Rn When q 0 = p, the sharp constant in (17) is −α −α/2 (19) Np,α,n = 2 π Γ Γ α/n n−α 2 Γ(n) n+α Γ n2 2 as computed in [L]. Inequality (17) expresses the sharp imbedding from 0 Lp (Rn ) ,→ L̇p−α (Rn ), where α = n(2/p − 1) > 0, 1 < p < 2, and L̇q−α (Rn ) = {f ∈ S 0 (Rn ) : kf kL̇q −α = kIα (|f |)kq < ∞} is a homogeneous Sobolev space. We now consider the case k = 3. When d1 = d2 = d3 = d and f1 = f2 = f3 = f , inequality (6) is a special case of a more general sharp inequality derived in [Mo]. In the special case 0 < d ≤ 2 it is possible to write Qd [f ] as a certain path integral, which is an L3 norm with an appropriate Wiener measure. This allows us to conclude that the expression 1/3 f → Qd [f ] = Qd,d,d [|f |, |f |, |f |] is a norm when 0 < d ≤ 2. It is quite natural to expect that Qd is a norm also when 2 < d < n, although we are not quite certain how to prove this in general. When d = 2n/3, however, it is an easy consequence of Theorem 1 that Qd is a norm since it is the L3 norm of a fractional integral. We have that for all f ≥ 0, Q2n/3 f = (2π)n/3 kIn/3 (f )k3 = (2π)n/3 kf kL̇3 Corollary 2. (20) . −n/3 To prove Corollary 2, take d1 = d2 = d3 = 2n/3 in (8). Then multiply (8) by f (x)f (y)f (z) and integrate with respect to dx, dy, and dz. Apply Fubini’s theorem and use (18) to obtain (20). Thus, in the special case d1 = d2 = d3 = 2n/3 inequality (6) is the same as inequality (17), with q = 3 = p0 and α = n/3. Observe that the constants also coincide since (|S n |−2 B(2n/3, 2n/3, 2n/3, n))1/3 = C(2n/3, 2n/3, 2n/3, n) = (2π)n/3 N3/2,n/3,n , as it should be. This gives a relationship between the sharp imbedding given by (6) and the sharp Sobolev imbedding given in (17) when d = 2n/3. It is fairly routine to check that the expression Qd f remains unchanged if f is replaced by (f ◦U )|JU |d/n , where U is a conformal transformation of Rn A SELBERG INTEGRAL FORMULA AND APPLICATIONS 91 and JU is its Jacobian. This transformation is scaled suitably to preserve Ln/d . We will denote by Bd the space of all measurable functions f on Rn such that Qd f < ∞. The main feature of the space Bd is the conformal invariance of its ‘norm’ Qd . It is reasonable to ask whether Bd is related to any L3 -based homogeneous Sobolev space. By homogeneity it can only be compared to L̇3n/3−d . We have the following: Theorem 2. The space Bd is contained in L̇3n/3−d when n > d > 2n/3 but Bd contains L̇3n/3−d when 2n/3 > d > n/3; furthermore both inclusions are strict. Quantitatively speaking, for any n > d > 2n/3 there exists a constant C = Cd,n such that for all measurable functions f we have (21) kId−n/3 (|f |)k3 ≤ CQd [f ]. For any 2n/3 > d > n/3 there exists a constant C = Cd,n such that for all measurable functions f we have (22) Qd [f ] ≤ CkId−n/3 (|f |)k3 . 4. The proof of Theorem 2. Observe that the cube of the left hand side of (21) is equal to Z Cd,n Kd (x, y, z)f (x)f (y)f (z)dxdydz, R3n where Z d− 4n d− 4n d− 4n 3 |y − t| 3 |z − t| 3 dt. Kd (x, y, z) = |x − t| Rn If we establish that for 2n/3 < d < 2n we have (23) Kd (x, y, z) ≤ Cd,n |x − y|d−n |y − z|d−n |z − x|d−n , then (21) will follow immediately. Similarly, if we prove that for n/3 < d < 2n/3 we have (24) |x − y|d−n |y − z|d−n |z − x|d−n ≤ Cd,n Kd (x, y, z), then (22) will follow as well. Now a simple dilation implies that (23) and (24) are valid for |x|, |y|, |z| ≤ 1, then they are valid for |x|, |y|, |z| ≤ R with the same constant for all R > 0. Letting R → ∞ we conclude that (23) and (24) are valid for all R > 0. Therefore, it suffices to prove (23) and (24) for |x|, |y|, |z| ≤ 1. Given any three points x, y, z in Rn , let M (x, y, z) = max(|x − y|, |y − z|, |z − x|) be their maximum and m(x, y, z) = min(|x − y|, |y − z|, |z − x|) be their minimum. Let us also call µ(x, y, z) the number in the middle. Then we have that µ(x, y, z) ≥ 21 M (x, y, z). The following lemma gives us asymptotic estimates for Kd (x, y, z). 92 Lemma. LOUKAS GRAFAKOS AND CARLO MORPURGO Let |x|, |y|, |z| ≤ 1. Then for 5n/6 < d < n we have Kd (x, y, z) ∼ M (x, y, z)3(d−n) . (25) For d = 5n/6 we have (26) n Kd (x, y, z) ∼ M (x, y, z)− 2 log M (x, y, z) , m(x, y, z) and for n/3 < d < 5n/6 we have (27) n Kd (x, y, z) ∼ m(x, y, z)−n+2(d− 3 ) M (x, y, z)d− 4n 3 . Now (23) and (24) are easy consequences of this lemma and of the observation that µ(x, y, z) is always comparable to M (x, y, z). Let us now give sketch the proof of the lemma above. Since the problem is translation invariant, it suffices to study the asymptotic behavior of the integral below as |α|, |β| → 0 Z (28) |α − t|−n+λ |β − t|−n+λ |t|−n+λ dt, n R where we set λ = d − n/3 and α = z − x and β = z − y. Since both |α|, |β| ≤ 2 the problem is local and we consider the following five cases: Case 1. |α| → 0, |β| ∼ |β − α| ∼ 1. Case 2. |α − β| → 0, |β| ∼ |α| ∼ 1. Case 3. |β − α| |α| ∼ |β| → 0. Case 4. |β − α| ∼ |α| ∼ |β| → 0. Case 5. |α| |β| ∼ |β − α| → 0. It is easy to see that in Case 1, the integral (28) behaves like a constant when λ > n/2, blows up like |α|−n+2λ when λ < n/2 and also blows up like log |α|−1 when λ = n/2. Case 2 is similar to Case 1 where the roles of |α| and |β − α| are interchanged. In Case 3 the situation is slightly different. The integral (28) behaves asymptotically like |α|−n+λ |α − β|−n+2λ when λ < n/2, as |α|−2n+3λ when λ > n/2, and as |α|−2n+3λ log(|α||α − β|−1 ) when λ = n/2. In Case 4, the integral (28) behaves asymptotically like |α − β|−2n+3λ . Finally, Case 5 follows from Case 1. In this case one has asymptotic behavior |α|−n+2λ |β|−n+λ when λ < n/2, |β|−2n+3λ when λ > n/2, and |β|−2n+3λ log(|β|/|α|) when λ = n/2. The derivation of the asymptotics of (28) in each case involves different splitting of the integral (28) and use of formula (12). The details are rather tedious and are omitted. A SELBERG INTEGRAL FORMULA AND APPLICATIONS 93 The exceptional case λ = n/2 corresponds to d = 5n/6 and only in this case a logarithmic term appears. We now indicate how the behavior of K(x, y, z) follows from the asymptotic behavior of the integral (28). First take 5n/6 < d < n, equivalently n/2 < λ < 2n/3. Recalling that |α| = |z − x| and |β| = |z − y|, we observe that the asymptotics in the five cases above, (i.e. C in Cases 1 and 2, |x − z|−2n+3λ in Case 3, |x − y|−2n+3λ in Case 4, and |y − z|−2n+3λ in Case 5) is a restatement of (25). Likewise, the statements in the five cases above can be summarized in (26) when d = 5n/6, and in (27) when n/3 < d < 5n/6. Using again the asymptotics for Kd , one can construct examples to show that the converse inequalities to (21) and (22) are false. The details are omitted. This concludes the proof of Theorem 2. 5. An application to fractional integrals. Formula (8) can be used to give an alternative proof of inequality (6) in the particular case d1 + d2 + d3 = 2n. Observe that in this case 1/p1 + 1/p2 + 1/p3 = 2 while 1/p01 + 1/p02 + 1/p03 = 1. Use (8) to rewrite Qd1 ,d2 ,d3 (f1 , f2 , f3 ) as Z 1 (29) (f1 ∗ | · |−d2 )(t) (f2 ∗ | · |−d3 )(t) (f3 ∗ | · |−d1 )(t) dt. B(d1 , d2 , d3 , n) Rn Apply Hölder’s inequality to estimate (29) by (30) 1 kf1 ∗ | · |−d2 kp01 kf2 ∗ | · |−d3 kp02 kf3 ∗ | · |−d1 kp03 . B(d1 , d2 , d3 , n) (17), (18), and (19) now imply that (30) is bounded by dj n−dj 3 Y dj Γ( 1 Γ(n/2) −1+ n 2 ) 2 π kfj kpj d B(d1 , d2 , d3 , n) Γ(n − 2j ) Γ(n) j=1 Q which is nothing else than C(d1 , d2 , d3 , n) 3j=1 kfj kpj . Acknowledgments. The authors would like to thank Bill Beckner for helpful comments and discussions. References [B] W. Beckner, Geometric inequalities in Fourier analysis, Essays on Fourier Analysis in honor of Elias M. Stein (Princeton, NJ, 1991), 36-68, Princeton Math. Ser. 42, eds. C. Fefferman, R. Fefferman and S. Wainger, Princeton University Press, Princeton, NJ, 1995. 94 LOUKAS GRAFAKOS AND CARLO MORPURGO [Ca] J.M. Caillol, Exact results for a two-dimensional one component plasma on a sphere, J. Phys. Lettres, 42 (1981), L245-L247. [C1] M. Christ, On the restriction of the Fourier transform to curves: Endpoint results and the degenerate case, Trans. Amer. Math. Soc., 287 (1985), 223-238. [C2] M. Christ, Estimates for the k-plane transform, Indiana Univ. Math. J., 33 (1984), 891-910. [D] S.W. Drury, Lp estimates for the x-ray transform, Illinois J. Math., 27 (1983), 125129. [GS] I.M. Gelfand and G.E. Shilov, Generalized functions, Vol. I, Academic Press, New York and London, 1964. [L] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1883), 349-374. [Me] M.L. Mehta, Random Matrices, 2nd ed., Academic Press, Boston, MA, 1990. [Mo] C. Morpurgo, Conformally invariant Sobolev inequalities and the spectrum of intertwinning operators, preprint, 1999. [Se] A. Selberg, Bemerkninger om et multiplet integral, Norsk Matematisk Tidsskrift, 26 (1944), 71-78. [St] E.M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ, 1970. Received February 3, 1998 and revised April 1, 1998. The research of the first author was supported by the NSF under grant DMS 9623120. The research of the second author was performed at the University of Texas at Austin, and supported by the NSF under grant DMS 9622891 Dipartimento di Matematica e Applicazioni Universita di Milano - Bicocca Via Bicocca degli Arcimboldi, 8 20126 Milano Italy Dipartimento di Matematica e Applicazioni Universita di Milano - Bicocca Via Bicocca degli Arcimboldi, 8 20126 Milano Italy E-mail address: [email protected] PACIFIC JOURNAL OF MATHEMATICS Vol. 191, No. 1, 1999 ASCENT AND DESCENT FOR FINITE SEQUENCES OF COMMUTING ENDOMORPHISMS Luzius Grunenfelder and Matjaž Omladič Homological techniques involving the Koszul complex are used to define and explore two invariants, ascent and descent, for a finite sequence of commuting endomorphism of a module. It is shown in particular that, as in the case of a single endomorphism, if ascent and descent are both finite then they are equal, and that this finiteness condition is equivalent to a certain strong Fitting type property. 1. Introduction. Let A be an algebra over a commutative ring R and let M be a left Amodule. If a : M → M is an A-endomorphism then ker ai ⊆ ker ai+1 and im ai ⊇ im ai+1 for every i ≥ 0. The ascent of a is the least positive integer r for which ker ar = ker ar+1 and the descent is the least positive integer s for which im as = im as+1 , if such integers exist and ∞ if they don’t. If both the ascent r and the descent s of a are finite then r = s and M = ker ar ⊕ im ar . This is Fitting’s Lemma. It holds in particular for every a ∈ EndA (M) if the A-module M is both Artinian and Noetherian. More generally, we may say that a has the Fitting property if M = K ⊕ I, where K = ∪r ker ar and I = ∩r ar M. In this paper we consider n-tuples a = (a1 , a2 , . . . , an ) of commuting endomorphisms of the A-module M. The ascent and the descent of a are defined and investigated by means of homological techniques. The role of the single endomorphism a in the classical case will be taken by the Koszul complex K(M, ∂a ) of the n-tuple a. Composition by the endomorphism a is replaced by the procedure of forming the diagonal complex of the double complex obtained via tensoring by the Koszul complex K(A, ∂a ), where A is a commutative subalgebra of EndA (M) containing the n-tuple a. This leads to a natural extension of the concepts of ascent, descent and of Fitting’s Lemma to finite sequences of commuting endomorphisms in Sections 2 to 5. The whole approach works for any module M over a commutative ring A and any finite sequence of elements of A, without specific reference to R and to an A-module structure on M. However, in our context the Fitting decomposition is of course A-invariant. In general, we say that the n-tuple a has the Fitting property if M = K ⊕ I and aI = I, where 95 96 L. GRUNENFELDER AND M. OMLADIČ K = ∪r HomA (A/ar , M), I = ∩s as M and a is the ideal in A generated by the n-tuple a. The main result of Section 5 is that a has finite ascent and finite descent if and only if a has the Fitting property and a acts nilpotently on K. In case R is a field, we show in Section 7 and 8 that the concepts of ascent and descent are actually functions defined on finite dimensional subspaces of commuting elements of EndA (M). More precisely, if the commuting ntuple a and the commuting m-tuple b span the same subspace in EndA (M) then they yield the same ascent, the same descent and the same Fitting decomposition of M. Localization techniques are then employed to see that these concepts are functions defined on finitely generated commutative subalgebras of EndA (M), i.e. if the finite sequences a and b generate the same ideal in the commutative subalgebra A of EndA (M) then they yield the same ascent and the same descent, even in the absence of the Fitting property. Apart from their independent interest, the results presented in this paper have also been motivated by some open problems in functional analysis and operator theory. It is folk knowledge in this theory that the spectral behaviour of a single compact operator in the vicinity of a non-zero spectral point can be studied using an analog of Fitting’s Lemma [TAE, p. 271ff]. It was J.S. Taylor [T] who introduced the Koszul complex into the discussion of the joint spectrum of a commuting n-tuple of bounded operators. The question of how to study the spectral behaviour of such an n-tuple is of considerable importance, in particular in applications like multiparameter spectral theory [R] and the theory of elementary operators [C]. In [BT] the Taylor spectrum was applied to study the spectral properties of a commuting n-tuple of compact operators by combining the ideas of Fitting’s lemma and the Koszul complex. Our approach provides a Fitting type decomposition for any point in the Taylor spectrum. To establish notation and for the convenience of the reader some of the well-known results concerning the homology of Koszul complexes [S] are reviewed in Section 2. 2. Preliminaries on Koszul Complexes. 2.1. Throughout this article let A be a fixed associative algebra with unit over the commutative ring R, and let M be a fixed left A-module. Furthermore, let A be a commutative subalgebra of EndA (M). For any n-tuple a = (a1 , a2 , . . . , an ) of elements of A we may construct the Koszul complexes K(A, a) and K(M, a) as follows. If e = (e1 , e2 , . . . , en ) is a basis of the free A-module An then the differential graded A-algebra K(A, a) consists of the exterior algebra Λ(An ) together with the differential of degree one ∂a : Λ(An ) → Λ(An ) ASCENT AND DESCENT FOR FINITE SEQUENCES 97 P defined by ∂a (x) = ni=1 ai ei ∧ x. The Koszul cochain complex of M is then the differential graded A-module K(M, a) = M ⊗A K(A, a). It is often useful to use the recursive definition K(A, a) = K(A, a0 ) ⊗A K(A, an ) of the Koszul complex, where a0 = (a1 , a2 , . . . , an−1 ) and where K(A, an ) is just the complex an : A → A. Proposition 2.2. Let X be any cochain complex of A-modules and let a be an element of A. Then there is a short exact sequence in cohomology 0 → H 1 (H p−1 (X) ⊗A K(A, a)) → H p (X ⊗A K(A, a)) → H 0 (H p (X) ⊗A K(A, a)) → 0 for each integer p ≥ 0. Proof. Considering K(A, a) as the complex a : K 0 (A, a) → K 1 (A, a) then we get the short exact sequence of cochain complexes 0 → (X ⊗A K 1 (A, a))p−1 → (X ⊗A K(A, a))p → (X ⊗A K 0 (A, a))p → 0 and the associated long exact sequence in cohomology 1⊗ a A H p−1 (X) ⊗A K 0 (A, a) −→ H p−1 (X) ⊗A K 1 (A, a) → H p (X ⊗A K(A, a)) 1⊗ a A → H p (X) ⊗A K 0 (A, a) −→ H p (X) ⊗A K 1 (A, a). The assertion is now established, by taking the kernel of 1 ⊗A a on the right a H 0 (H p (X) ⊗A K(A, a)) = ker(1 ⊗A a) = ker(H p−1 (X) → H p−1 (X)) and of 1 p−1 course its cokernel on the left H (H (X) ⊗A K(A, a)) = coker(1 ⊗A a) = p−1 p−1 H (X)/aH (X). Corollary 2.3. Let X be any cochain complex of A-modules and let a be an element of A. If a is invertible then H ∗ (X ⊗A K(A, a)) = 0 and if a = 0 then H p (X ⊗A K(A, a)) ∼ = H p−1 (X) ⊕ H p (X). Corollary 2.4. If M = 6 0 is Artinian and Noetherian as an A-module and if a ⊂ rad A then the H p (K(M, a)) 6= 0 for 0 ≤ p ≤ n. Proof. Let a = (a1 , a2 , . . . , an ) be contained in the Jacobson radical rad A. The result will be established by induction on n. For n = 1 the exact sequence a 0 → H 0 (M, a) → M → M → H 1 (M, a) → 0 of Artinian and Noetherian A-modules describes the situation. If H 0 (M, a) = 0 then the map a : M → M is injective. The descending chain of submodules M ⊇ im a ⊇ im a2 ⊇ . . . ⊇ im aj ⊇ . . . must become stationary after finitely many steps since M is Artinian. But, im aj = im aj+1 = a(im aj ) means that aj (im a) = aj (M) so that aM = im a = M, since aj is injective. But if H 1 (M, a) = M/aM = 0 then 98 L. GRUNENFELDER AND M. OMLADIČ M = 0 by Nakayama’s Lemma, hence a contradiction. Thus, the case n = 1 is established. Now assume that n > 1, a0 = (a1 , a2 , . . . , an−1 ). The induction hypothesis is that the A-modules H q (M, a0 ) 6= 0 for all q satisfying 0 ≤ q ≤ n − 1 and that they are both Artinian and Noetherian. If H p (M, a) = 0 then by the exact sequences of Proposition 2.2 0 → H 1 (H p−1 (X) ⊗A K(A, a)) → H p (X ⊗A K(A, a)) → H 0 (H p (X) ⊗A K(A, a)) → 0 with X = K(M, a0 ) and a = an , we see that H p−1 (X)/an H p−1 (X) = 0 and ker(an : H p (X) → H p (X)) = 0. For p > 0, Nakayama’s Lemma implies that H p−1 (M, a0 ) = 0, which contradicts the induction hypothesis unless p > n. If p = 0, then H 0 (M, a) = ker(an : H 0 (M, a0 ) → H 0 (M, a0 )) = 0 so that an : H 0 (M, a0 ) → H 0 (M, a0 ) is an injective endomorphism of the Artinian A-module H 0 (M, a0 ), hence bijective. Again Nakayama’s Lemma implies that H 0 (M, a0 ) = 0 in contradiction to the induction hypothesis. Corollary 2.5. For any A-module M the commuting n-tuple a is in the annihilator of H ∗ (K(M, a)). Proof. This follows easily by induction on n using the exact sequence of Proposition 2.2 with X = K(M, a0 ) and a = an . 2.6. If X is a cochain complex of A-modules then X ⊗A K(A, a) becomes a bicomplex with differentials ∂ ⊗A 1 and 1 ⊗A ∂a . The diagonal complex D = D(X ⊗A K(A, a)) with Di = Xi ⊗A Ki has differential ∂ ⊗A ∂a . Let K (0) = A = D(0) and for r ≥ 1 define inductively the iterated diagonal complex D(r) = D(D(r−1) ⊗A K(A, a)), which is a differential graded A(r) algebra with Di = ⊗rA Ki (A, a) and differential ∂ (r) = ⊗rA ∂a = ∂ 1 ∂ 2 . . . ∂ r , where ∂ j : ⊗rA Ki (A, a) → ⊗rA Ki+1 (A, a) is acting like ∂a on the j-th and like the identity on the remaining tensor factors. Finally we introduce the differential graded D(r) A-module M(r) = M ⊗A D(r) . Since there is no danger of confusion, we shall use the same notation ∂ j and ∂ (r) for the corresponding differentials on both D(r) and M(r) . From the degree-wise and coordinate-wise point of view we may introduce an additional upper index on the basis elements indicating the factor in the (r) (r) tensor power Di = ⊗rA Ki (A, a). A basis of Dp is given by {e1I1 ⊗A e2I2 ⊗A r . . .⊗A eIr }, where eI = ei1 ∧ei2 ∧. . .∧eip for the multi-index I = (i1 , i2 , . . . , ip ) (r) (r) of type p with 1 ≤ i1 < i2 < · · · < ip ≤ n. Notice that Mi = M ⊗A Di (r) (r) (r) (r) and that ∂i : Mi → Mi+1 is the composite ∂i = ∂i1 ∂i2 . . . ∂ir , where P (r) (r) ∂ij x = ( nk=1 ak ejk ) ∧ x for every x ∈ Mi . We can also define Mi and (r) the differentials ∂i inductively using the commutative diagram in Figure 1. Observe that the main diagonal of this commutative diagram represents the ASCENT AND DESCENT FOR FINITE SEQUENCES 99 left most vertical of the diagram on the next step, i.e. with r replaced by r + 1. Every row and every column of the diagram represents a cochain complex. Also, the left most and the right most columns are actually isomorphic. Since Dj is a free A-module, the cohomology of the j-th column is H ∗ (M(r) ⊗A Dj ) ∼ = H ∗ (M(r) ) ⊗A Dj . The i-th row is of course isomorphic to the Koszul complex (r) (r) K(Mi , a) ∼ = K(M, a) ⊗A Di (r) of the A-module Mi cohomology is (r) (r) ∼ = M ⊗A Di and, since Di is a free A-module, its (r) (r) H ∗ (K(Mi ⊗A D, a)) ∼ = H ∗ (K(M, a)) ⊗A Di . In particular, if the A-module M is both Artinian and Noetherian and if (r) a ⊂ rad A, then it follows from Corollary 2.4 that H p (K(Mi ⊗A D)) 6= 0 for 0 ≤ p ≤ n. (r) r+1 ∂n−1 (r) ⊗A Dn (r) y∂0 (r) r+1 ∂n−1 (r) ⊗A Dn (r) y∂1 (r) r+1 ∂n−1 (r) (r) ∂ r+1 (r) ⊗A D1 (r) y∂0 ... M0 ⊗A Dn−1 −−−→ M0 (r) y∂0 (r) ∂ r+1 (r) ⊗A D1 (r) y∂1 ... M1 ⊗A Dn−1 −−−→ M1 (r) y∂1 −−0−→ M2 ⊗A D1 ∂ r+1 (r) ... M2 ⊗A Dn−1 −−−→ M2 ⊗A Dn M0 −−0−→ M0 (r) y∂0 M1 −−0−→ M1 (r) y∂1 (r) M2 .. . (r) .. . ∂ r+1 (r) Mn−1 −−0−→ Mn−1 ⊗A D1 (r) (r) ∂ y n−1 y∂n−1 (r) Mn ∂ r+1 (r) −−0−→ Mn ⊗A D1 .. . (r) .. . r+1 ∂n−1 (r) . . . Mn−1 ⊗A Dn−1 −−−→ Mn−1 ⊗A Dn (r) (r) ∂ y n−1 y∂n−1 ... (r) r+1 ∂n−1 (r) Mn ⊗A Dn−1 −−−→ Mn ⊗A Dn Figure 1. The composite of any of either a horizontal or a vertical by a consecutive diagonal map is zero and so is the composite of any diagonal by a consecutive horizontal or vertical map. Therefore, any path which changes 100 L. GRUNENFELDER AND M. OMLADIČ from horizontal to vertical direction or vice versa only through a diagonal is a cochain complex. Of particular interest are “Yoneda composites” of columns by rows and of rows by columns. 3. Some Properties of the Double Complexes. (r) At each point Mi ⊗A Dj of the double complex in Fig. 1, i.e. at that point in the commutative diagram (r) r+1 ∂j−1 (r) ∂jr+1 (r) (r) r+1 ∂j−1 (r) ∂jr+1 (r) (r) r+1 ∂j−1 (r) ∂jr+1 (r) Mi−1 ⊗A Dj−1 −−−→ Mi−1 ⊗A Dj −−−→ Mi−1 ⊗A Dj+1 (r) (r) (r) ∂i−1 y ∂i−1 y ∂i−1 y Mi ⊗A Dj−1 −−−→ Mi ⊗A Dj −−−→ Mi ⊗A Dj+1 (r) (r) (r) ∂i y ∂i y ∂i y Mi+1 ⊗A Dj−1 −−−→ Mi+1 ⊗A Dj −−−→ Mi+1 ⊗A Dj+1 Figure 2. we may consider the vertical, the vertical-diagonal and the diagonal-vertical (r) (r) (r) cohomologies, denoted by Hij , Lij and Rij , respectively. It follows in particular that (r) r H00 = ker ∂0 ∼ = Hom(A/ar , M) , (r) (r+1) (r) (r+1) L00 = H00 and Rnn = Hnn generated by the n-tuple a. (r) (r) Hnn = M/ im ∂n−1 ∼ = M/ar M, for all r ≥ 1, where a is the ideal in A Proposition 3.1. Let a be an n-tuple of endomorphisms contained in the commutative subalgebra A of EndA (M). (r) (r) (a) Suppose 0 ≤ j < n. If Llj = Hlj for some l, where 0 ≤ l < n, then (r) (r) Lkj = Hkj for all k, where 0 ≤ k ≤ l. (r) (r) (b) Suppose 0 < j ≤ n. If Rlj = Hlj for some l, where 0 < l ≤ n, then (r) (r) Rkj = Hkj for all k, where l ≤ k ≤ n. Proof. a) Observe that, due to the fact that the vertical and the verticaldiagonal cohomologies at a point are defined by the same incoming homomorphism, we only have to show that the outgoing maps have equal kernels. (r) However, since the diagonal map ∂jr+1 ∂k is a composite by the vertical map (r) (r) (r) ∂k , it suffices to prove that ker(∂jr+1 ∂k ) ⊆ ker(∂k ). In order to see this ASCENT AND DESCENT FOR FINITE SEQUENCES 101 (r) pick an element x ∈ Mk ⊗A Dj from the kernel of the outgoing diagonal map and write X x= vI1 I2 ...Ir Ir+1 eI1 ⊗ eI2 ⊗ . . . ⊗ eIr ⊗ eIr+1 . I1 ,I2 ,... ,Ir+1 Here every Ii for 1 ≤ i ≤ r is a multi-index of type (j1 , j2 , . . . , jk ), where 1 ≤ j1 < j2 < . . . < jk ≤ n, and the corresponding eIi = ej1 ∧ ej2 ∧ . . . ∧ ejk . Moreover, Ir+1 is a multi-index of type (h1 , h2 , . . . , hj ) with 1 ≤ h1 < h2 < . . . < hj ≤ n and eIr+1 = eh1 ∧ eh2 ∧ . . . ∧ ehj . This implies that X X (r) ap1 ap2 . . . apr vI1 I2 ...Ir Ir+1 ∂k x = I1 ,I2 ,... ,Ir+1 p1 ,p2 ,... ,pr (ep1 ∧ eI1 ) ⊗ (ep2 ∧ eI2 ) ⊗ . . . ⊗ (epr ∧ eIr ) ⊗ eIr+1 . By assumption we have X (r) 0 = ∂jr+1 ∂k x = X ap1 ap2 . . . apr apr+1 vI1 I2 ...Ir Ir+1 I1 ,I2 ,... ,Ir+1 p1 ,p2 ,... ,pr+1 (ep1 ∧ eI1 ) ⊗ (ep2 ∧ eI2 ) ⊗ . . . ⊗ (epr ∧ eIr ) ⊗ (epr+1 ∧ eIr+1 ). For i = 1, 2, . . . , r choose any multi-index Ji of type (j1 , j2 , . . . , jl−k ), where 1 ≤ j1 < j2 < . . . < jl−k ≤ n, and define eJi = ej1 ∧ ej2 ∧ . . . ∧ ejl−k . Letting e = eJ1 ⊗ eJ2 ⊗ . . . ⊗ eJr , we see that x ∧ e belongs to Mrl ⊗A Dj . Now (r) (r) ∂jr+1 ∂l (x ∧ e) = (∂jr+1 ∂l x) ∧ e, since the diagram (r) ∂ (r) ∂ (r) (r) ∂jr+1 (r) (r) ∂jr+1 (r) Mk ⊗A Dj −−k−→ Mk+1 ⊗A Dj −−−→ Mk+1 ⊗A Dj+1 ∧ey ∧ey ∧ey (r) Ml ⊗A Dj −−l−→ Ml+1 ⊗A Dj −−−→ Ml+1 ⊗A Dj+1 (r) (r) is commutative. Thus, x ∈ ker(∂jr+1 ∂k ) implies that x ∧ e ∈ ker(∂jr+1 ∂l ) (r) and by the assumptions in part a) also that x ∧ e ∈ ker ∂l for every e as (r) (r) described above. But then (∂k x) ∧ e = ∂l (x ∧ e) = 0 for every such e. Clearly there are sufficiently many maps ∧e of that kind to conclude that (r) ∂k x = 0. b) Similar arguments apply to the second part of the Proposition. First observe that at a given point both the vertical and the diagonal-vertical cohomologies are defined by the same outgoing homomorphism. It therefore suffices to prove that the ingoing maps have equal images. However, since the (r) (r) r is a composite by the vertical map ∂k−1 , it remains diagonal map ∂k−1 ∂j−1 (r) (r) (r) r ). To this effect pick x ∈ M to show that im(∂k−1 ) ⊆ im(∂k−1 ∂j−1 k−1 ⊗A Dj 102 L. GRUNENFELDER AND M. OMLADIČ and write X x= vI1 I2 ...Ir Ir+1 eI1 ⊗ eI2 ⊗ . . . ⊗ eIr ⊗ eIr+1 . I1 ,I2 ,... ,Ir+1 Here every Ii for 1 ≤ i ≤ r is a multi-index of type (j1 , j2 , . . . , jk−1 ), where 1 ≤ j1 < j2 < . . . < jk−1 ≤ n, and the corresponding eIi = eij1 ∧ eij2 ∧ . . . ∧ eijk−1 . Moreover, Ir+1 is a multi-index of type (h1 , h2 , . . . , hj ) with 1 ≤ h1 < h2 < . . . < hj ≤ n and eIr+1 = eh1 ∧ eh2 ∧ . . . ∧ ehj . This implies that X X (r) ap1 ap2 . . . apr vI1 I2 ...Ir Ir+1 u = ∂k−1 x = I1 ,I2 ,... ,Ir+1 p1 ,p2 ,... ,pr (ep1 ∧ eI1 ) ⊗ (ep2 ∧ eI2 ) ⊗ . . . ⊗ (epr ∧ eIr ) ⊗ eIr+1 . For i = 1, 2, . . . , r choose any multi-indices Ji of type (j1 , j2 , . . . , jl−k ), where 1 ≤ j1 < j2 < . . . < jk−l ≤ n, in such a way that Ji is a subindex of Ii , i.e. such that every index that belongs to Ji also belongs to Ii . Next, let eJi = eij1 ∧ eij2 ∧ . . . ∧ eijk−l . Then e = eJ1 ⊗ eJ2 ⊗ . . . ⊗ eJr defines a (r) (r) homomorphism ∧e : Ms ⊗A Dj → Ms+k−l ⊗A Dj for each s so that the diagram r+1 ∂j−1 (r) Ml−1 ⊗A Dj−1 −−−→ ∧ey r+1 ∂j−1 (r) (r) ∂l−1 (r) Ml−1 (r) ⊗A Dj −−−→ Ml ⊗A Dj ∧ey ∧ey (r) ∂k−1 (r) (r) Mk−1 ⊗A Dj−1 −−−→ Mk−1 ⊗A Dj −−−→ Mk ⊗A Dj is commutative. According to the construction of e there must be an element (r) y ∈ Ml−1 ⊗A Dj such that x = y ∧ e. By the assumption of the Proposition (r) (r) (r) (r) r+1 that Rlj = Hlj we have ∂l−1 y ∈ im(∂l−1 ∂j−1 ), and there exists a z ∈ (r) (r) (r) r+1 Ml−1 ⊗A Dj−1 such that ∂l−1 ∂j−1 z = ∂l−1 y. By the commutativity of the (r) (r) (r) r+1 r+1 diagram we see that ∂k−1 ∂j−1 (z ∧ e) = (∂l−1 ∂j−1 z) ∧ e = (∂l−1 y) ∧ e = (r) (r) ∂k−1 (y ∧ e) = ∂k−1 x, proving the assertion. Corollary 3.2. Let a be an n-tuple of endomorphisms contained in the commutative subalgebra A of EndA (M). (r) (r) (r) (r) (a) If Ll0 = Hl0 for some l, 0 ≤ l < n, then Lk0 = Hk0 for all k, 0 ≤ k ≤ l. (r) (r) (r) (r) (b) If Rln = Hln for some l, 0 < l ≤ n, then Rkn = Hkn for all k, l ≤ k ≤ n. 3.3. Example 5.5 will show that the opposite of this Corollary is not true in general. However, we will also show that in some cases, and in particular in ASCENT AND DESCENT FOR FINITE SEQUENCES 103 the case of an A-module M which is both Artinian and Noetherian, equality (r) of the vertical and the vertical-diagonal cohomology modules at M0 , i.e. (r) at the top left corner of Fig. 1, implies their equality at Mn−1 . Thus, this (r) condition imposed on the cohomology modules at some point Mk , where 0 ≤ k < n, implies that the same condition holds for all k. Similar remarks apply to the case of the cohomology modules on the right-most vertical of Fig. 1. 4. Ascent and Descent. In the following we shall be mainly interested in the cohomology modules at the left most and the right most verticals of the double complexes M(r) ⊗A D, (r) that is in the situation of Corollary 3.2. Throughout the modules Mn and (r) M0 will be identified with M and a is the ideal in A generated by a. (r) (r) Proposition 4.1. If Ln−1,0 = Hn−1,0 for some index r ≥ 1, then (r) (r) (a) ker(∂0r+l . . . ∂0r+1 ∂n−1 ) = ker(∂n−1 ) for all l ≥ 1; (r) (r) (b) ker ∂0 ∩ im ∂n−1 = 0, i.e. HomA (A/ar , M) ∩ ar M = 0; (s) (s) (c) Ln−1,0 = Hn−1.0 for all indices s ≥ r. Proof. (a) The case l = 1 is just our assumption. Assume the claim true for (r) (r) a certain index l − 1 and observe that ker(∂0r+l . . . ∂0r+1 ∂n−1 ) ⊇ ker ∂n−1 , so that we only have to show that in fact the opposite inclusion holds. To this (r) (r) end pick x ∈ Mn−1 such that ∂0r+l . . . ∂0r+1 ∂n−1 x = 0 and write y = ∂0r+l−1 . . . ∂0r+1 x = X r+2 r+l−1 xi1 i2 ...il−1 er+1 i1 ⊗ ei2 ⊗ . . . ⊗ eil−1 . i1 ,i2 ,... ,il−1 (r) Clearly, xi1 i2 ...il−1 ∈ Mn−1 , and since ∂0i commutes with ∂0j for i 6= j we (r) (r) see that ∂0r+l ∂n−1 y = 0. This forces ∂n−1 y = 0 by the assumption in the (r) statement of our proposition. Since the maps ∂0i commute with ∂n−1 , we (r) conclude that ∂0r+l−1 . . . ∂0r+1 ∂n−1 x = 0 and the assertion follows by the induction hypothesis. (r) i (b) The homomorphism ∂n−1 is a composite of maps of the type ∂n−1 (r) for 1 ≤ i ≤ r. After identifying M0 (r) (r) with Mn−1 we may write ∂0 = (r) (r) (r) If x ∈ ker ∂0 ∩ im ∂n−1 then by assumption x = ∂n−1 y (r) (r) for some y ∈ Mn−1 and ∂02r ∂02r−1 . . . ∂0r+1 x = ∂0 x = 0. Consequently (r) (r) ∂02r ∂02r−1 . . . ∂0r+1 ∂n−1 y = 0 and it follows by (a) that x = ∂n−1 y = 0. ∂02r ∂02r−1 . . . ∂0r+1 . 104 L. GRUNENFELDER AND M. OMLADIČ (r+1) (c) Choose any x ∈ ker(∂n−1 ∂0r+2 ) and observe that it suffices to show (r+1) (r+1) (r) r+1 that x ∈ ker ∂n−1 . Recall that ∂n−1 = ∂n−1 ∂n−1 and that the two homo(r) r+1 morphisms commute. We see that y = ∂n−1 x ∈ ker(∂n−1 ∂0r+2 ) and hence (r) (r) r+1 by assumption y ∈ ker ∂n−1 . This finally shows that 0 = ∂n−1 ∂n−1 x = (r+1) ∂n−1 x. (r) (r) Proposition 4.2. If R1n = H1n for some index r ≥ 1, then (r) (r) r+l r+1 (a) im(∂n−1 . . . ∂n−1 )∂0 = im ∂0 (b) (c) (r) (r) ker ∂0 + im ∂n−1 = M, i.e. (s) (s) R1n = H1n for all indices s for all l ≥ 1; HomA (A/ar , M) + ar M = M; ≥ r. Proof. (a) The case l = 1 is identical with our assumption. Suppose that r+1 (r) r+l )∂0 ⊆ . . . ∂n−1 the claim is true for an index l − 1 and observe that im(∂n−1 (r) im ∂0 , so that it suffices to show that the opposite inclusion holds. To this (r) (r) end pick x ∈ M1 ⊗A Dn such that x ∈ im ∂0 and assume by the induction (r) r+2 (r) r+l )∂0 y for some y ∈ M0 ⊗l−1 . . . ∂n−1 hypothesis that x = (∂n−1 A Dn−1 . Then X r+3 r+l y= yi2 i3 ...il êr+2 , i2 ⊗A êi3 ⊗A · · · ⊗A êil i2 ,i3 ,... ,il where the exterior products êji = ∧p6=i ejp form the canonical basis elements (r) (r) (r) of Dn−1 and where yi2 i3 ...il ∈ M0 ∼ = M0 ⊗A Dn . Clearly, ∂0 yi2 i3 ...il ∈ (r) M1 ⊗A Dn , so that by the assumption in our proposition there is an element (r) (r) r+1 (r) zi2 i3 ...il ∈ M0 ⊗A Dn−1 such that ∂0 yi2 i3 ...il = ∂n−1 ∂0 zi2 i3 ...il . Since the j i maps ∂n−1 and ∂n−1 commute for i 6= j, we conclude for X r+3 r+l z= zi2 i3 ...il êr+2 i1 ⊗A êi2 ⊗A . . . ⊗A êil i2 ,i3 ,... ,il (r) r+l r+1 that (∂n−1 . . . ∂n−1 )∂0 z = x, which proves the assertion. (r) (b) Choose any x ∈ M. The homomorphism ∂0 is a composite of maps (r) (r) of the type ∂0i for 1 ≤ i ≤ r. After identifying M0 with Mn we may (r) 2r ∂ 2r−1 . . . ∂ r+1 . By Corollary 3.2. (b) the element ∂ (r) x write ∂n−1 = ∂n−1 n−1 n−1 0 (r) (r) belongs to im(∂n−1 ∂0 ), yielding the existence of an element y for which (r) (r) (r) (r) (r) ∂n−1 ∂0 y = ∂0 x and hence ∂0 (x − ∂n−1 y) = 0. (r+1) (c) Pick any x ∈ im ∂0 and observe that it obviously suffices to show (r+1) r+2 (r+1) (r) that x ∈ im(∂0 ∂n−1 ). Recall first that ∂0 = ∂0r+1 ∂0 and that the (r) (r) two maps commute. If y is such that x = ∂0r+1 ∂0 y then ∂0r+1 y ∈ im ∂0 = ASCENT AND DESCENT FOR FINITE SEQUENCES 105 (r) r+2 im(∂n−1 ∂0 ) by the induction hypothesis and by the assumption in the (r) (r+1) r+2 Proposition. Thus we conclude that x = ∂0 ∂0r+1 y ∈ im(∂n−1 ∂0 ). (r) 4.3. If there exists an index r for which at the point Mn−1 ⊗A D0 the (r) left vertical cohomology Hn−1,0 is equal to the vertical-diagonal cohomology (r) Ln−1,0 , then by Proposition 4.1 the same is true for all larger indices. The smallest index r with this property will be called the ascent of the n-tuple a. If such an index does not exist, we shall say that the ascent of the n-tuple is infinite. (s) Similarly, if there is an index s for which at the point M1 ⊗A Dn the right (s) (s) vertical cohomology H1n is equal to the diagonal-vertical cohomology R1n , then by Proposition 4.2 the same holds for all larger indices. The smallest index s with this property will be called the descent of the n-tuple a. If such an index does not exist, we shall say that the descent of the n-tuple is infinite. More generally, for any pair (i, j) satisfying 0 ≤ i, j ≤ n, let d0ij be the (r) (r) smallest among the indices r for which Li,j = Hi,j and d1ij the smallest (r) (r) among the indices r for which Rij = Hij . We let any of these indices be infinite whenever there is no r satisfying the respective condition. According to this notation the ascent of the n-tuple a is d00,n−1 and the descent is d1n1 . 5. Relations between ascent, descent and Fitting’s Lemma. In this section we will show that both ascent and descent of a commuting n-tuple a are finite if and only if a has the Fitting property and a acts nilpotently on K, and that ascent and descent are equal in this case. Let us recall from Section 4, that the notation for the ascent and the descent are d0 = d00,n−1 and d1 = d11n , respectively. It is also useful to remember that (r) (r) (r) (r+1) (r) (r+1) H ∼ and Rnn = Hnn . = HomA (A/ar , M), Hnn ∼ = M/ar M, L = H 00 00 00 Proposition 5.1. Let a be an n-tuple of endomorphisms contained in the commutative subalgebra A of EndA (M). (r) (r) (s) (s) (a) If L00 = H00 for some index r ≥ 1, then L00 = H00 for all s ≥ r. (s) (s) (r) (r) (b) If Rnn = Hnn for some index r ≥ 1, then Rnn = Hnn for all s ≥ r. (r) (r) (s) (s) (c) If L00 = H00 for a fixed index r ≥ 1 and R1n = H1n for some index s ≥ 1, then this holds for some s ≤ r. (r) (r) (s) (s) (d) If Rnn = Hnn for a fixed index r ≥ 1 and Ln−1,0 = Hn−1,0 for some index s ≥ 1 then this holds for some s ≤ r. Proof. For (a) and (b) use similar arguments as in the proofs of Propositions 4.1(a) and 4.2(a). 106 L. GRUNENFELDER AND M. OMLADIČ (s) (s) (c) Let s be the smallest index for which R1n = H1n and suppose that (r) (r) the assertion is false. Then, there must be an r < s with L00 = H00 . By (a) we may assume with no loss of generality that r = s − 1. We will (s−1) (s−1) show that in this case we must also have R1n = H1n , in contradiction to the minimality of s, thus proving the assertion. Notice that we always s ∂ (s−1) ) ⊆ im ∂ (s−1) at the point M(s) ⊗ D , so that only have im(∂n−1 n A 0 0 1 the opposite inclusion has to be established. For this purpose pick any (s−1) (s−1) (s−1) x ∈ im ∂0 . Then x = ∂0 y for some y ∈ M0 ⊗A Dn and it follows (s) (s) s+1 s that ∂0 x ∈ im ∂0 = im ∂0 ∂n−1 by the second assumption in (c). This (s) (s) yields the existence of an element z ∈ M0 ⊗A Dn−1 such that ∂(0) y = ∂0s x = (s) (s) (s) (s−1) (s−1) s+1 s+1 ∂0 ∂n−1 z, and hence y − ∂n−1 z ∈ ker ∂0 . Thus H00 = L00 = H00 , where the first equality holds by definition and the second by assumption. (s−1) (s−1) s+1 (s−1) s+1 ∂n−1 ), Therefore, x = ∂0 y = ∂0 ∂n−1 z which belongs to im(∂0 (s−1) (s−1) s+1 ) and hence the required establishing the inclusion im ∂0 ⊆ im(∂0 ∂n−1 contradiction. (s) (s) (d) Let s be the smallest index for which Ln−1,0 = Hn−1,0 and suppose (r) that the assertion is false. Then, there must be an r < s such that Rnn = (r) Hnn . By (b) we may assume with no loss of generality that r = s − 1. We (s−1) (s−1) shall show that in this case we must also have Ln−1,0 = Hn−1,0 , contradicting the minimality of s and therefore proving our assertion. Recall that we (s−1) (s−1) always have ker ∂n−1 ⊆ ker(∂n−1 ∂0s ), so that only the opposite inclusion (s−1) (s−1) has to be established. Pick any x ∈ ker(∂n−1 ∂0s ), then y = ∂n−1 x ∈ (s−1) (s) (s−1) ker ∂0s ∩ im ∂n−1 . Since Hnn = Rnn (s−1) = Hnn , where the first equality (s) holds by definition and the second by assumption, it follows that y = ∂n−1 z (s) (s) for some z ∈ Mn−1 and hence 0 = ∂0s y = ∂0s ∂n−1 z. This implies, by the (s) (s) (s) (s−1) assumption Ln−1,0 = Hn−1,0 , that y = ∂n−1 z = 0 and hence x ∈ ker ∂n−1 , establishing the required contradiction. Theorem 5.2. The ascent d0 and the descent d1 of a commuting n-tuple a are both finite if and only if M = K ⊕ I as an A-module, where the ideal a generated by the n-tuple a acts nilpotently on K, HomA (A/a, I) = 0 and aI = I. If these equivalent conditions are satisfied, then (a) d0 = r = d1 . (b) d0j−1,0 = r = d1jn for 1 ≤ r ≤ n. (r) (r) (c) K = ker ∂0 ∼ = HomA (A/ar , M) and I = im ∂n−1 = ar M. ASCENT AND DESCENT FOR FINITE SEQUENCES (r) (d) The restrictions of the ai to ker ∂0 HomA (A/a(k) , I) = 0 107 are nilpotent. Moreover, and a(k) I = I for every k ≥ 1, where a(k) is the ideal in A generated by the n-tuple a(k) = (ak1 , ak2 , . . . , akn ). (e) r is an invariant of the ideal a, i.e. independent of the particular finite generating set a. Proof. Recall that the ascent is d0 = d0n−1,0 and the descent d1 = d11n . Suppose that they are both finite. By Propositions 4.1(c) and 5.1(c) we have that d0n−1,0 ≥ d0n−2,0 ≥ . . . ≥ d00,0 ≥ d11n , proving in particular that the ascent is no smaller than the descent. On the other hand, Propositions 4.2(c) and 5.1(d) yield d11n ≥ d12n ≥ . . . ≥ d1nn ≥ d0n−1,0 and, in particular, that the descent is no smaller than the ascent. Thus all the indices and, in particular the ascent and the descent, are equal as required in (b) and in (a). Assertion (c) is an easy consequence of Propositions 4.1(b) and 4.2(b). That the decomposition is invariant under the actions of A and of A is due to the fact that the differentials are compatible with both actions. Now, (d) follows directly from (c). Conversely, suppose that M = K ⊕ I, where the restriction of the ai to K are nilpotent and the restrictions a0i of the ai to I satisfy ∩ni=1 ker a0i =0 and n X im a0i = I. i=1 Then for each r ≥ 1 there is obviously an isomorphism of complexes ∼ K(r) ⊕ I (r) , M(r) = and so it suffices to prove the assertion separately for K and for I. For N equal to K or I, as the case may be, consider first the diagram (r) Nn−1 (r) ∂n−1 y (r) ∂ r+1 (r) Nn −−n−→ Nn ⊗A D1 . Let Ei = e1 ∧ . . . ∧ êi ∧ . . . en . Then for N = K, we see that X (r) xi1 i2 ...ir Ei1 ⊗ Ei2 ⊗ . . . ⊗ Eir ∂n−1 i1 ,i2 ,... ,ir = X ai1 ai2 . . . air xi1 i2 ...ir = 0 i1 ,i2 ,... ,ir if r is large enough, since each coefficient xi1 i2 ...ir is in K. It suffices to choose P (r) r ≥ i ri , where ari i K = 0. Thus, there is an index r0 , such that ∂n−1 = 0 108 L. GRUNENFELDER AND M. OMLADIČ for all r ≥ r0 , and (a, K) has finite ascent. On the other hand, when N = I then X ∂nr+1 (y) = a0j y ⊗ ej = 0 j if and only if y = 0, since y ∈ I and ∩j ker a0j = 0. Hence, ∂nr+1 is injective for every r and (a0 , I) has finite ascent. Now consider the diagram (r) N0 r+1 ∂n−1 ⊗A Dn−1 −−−→ N (r) ⊗A Dn (r) y∂0 (r) N1 ⊗A Dn . Then, setting N = K, we see that X (r) ∂0 (x) = aj1 aj2 . . . ajr xej1 ⊗ ej2 ⊗ . . . ⊗ ejr = 0 ji ,j2 ,... ,jn P if r is large enough, since x ∈ K. Again, it suffices to choose r ≥ i ri , (r) where ari i K = 0. Thus, there is an index s0 such that ∂0 = 0 for all r ≥ s0 , so that (a, K) has finite descent. If N = I then ! X X r+1 xi e1 ∧ . . . ∧ êi ∧ . . . ∧ en = (−1)i−1 a0i xi . ∂n−1 i i r+1 ∂n−1 P is surjective for every r, since I = i im a0i , so that (a0 , I) Thus, has finite descent. The ascent and the descent of a are therefore both finite, hence equal. Moreover, it follows directly from our conditions, which are independent of the particular finite generating set of the ideal a, that this number r is the least positive integer for which one and hence all the conditions ar K = 0, I = ar M and K = HomA (A/ar , M) are satisfied, so that (e) holds. 5.3. The conditions HomA (A/a, I) = 0 and I = aI say that the n-tuple a acts ‘jointly bijectively’ on I. Observe that a slight modification of the proof of Theorem 5.2 shows that ascent and descent are either both finite or both infinite whenever a has the Fitting property, i.e. whenever M = K ⊕ I and I = aI, where K = ∪r Homa (A/ar , M) and I = ∩r ar M. As a more general version of assertion (e) we shall see in Section 8 that, even in the absence of the Fitting property, both ascent and descent are invariants of the ideal a. We say that the commuting n-tuple satisfies condition (F) if the A-module M decomposes into a direct sum of A-submodules M = K ⊕ I1 ⊕ I2 ⊕ . . . ⊕ Ip , ASCENT AND DESCENT FOR FINITE SEQUENCES 109 where Ii for 0 ≤ i ≤ p are invariant under A, where the restrictions of the aj to K are nilpotent, while for i = 1, 2, . . . , p at least one of the restrictions to Ii is invertible. Observe that if M is both Artinian and Noetherian as an A-module, then every commuting n-tuple of endomorphisms satisfies this condition. Indeed, in this case the ascent and the descent of the endomorphism a1 must be finite by Fitting’s Lemma [P, B]. By Theorem 5.2, applied to the 1-tuple a1 and the fixed commutative algebra A, the A-module M decomposes into M = K1 ⊕ I1 , where K1 = ker ar1 and I1 = im ar1 for some sufficiently large exponent r. By commutativity, K1 and I1 are A-invariant. If the restrictions of all the ai to K1 are nilpotent, then we are done. If not, pick one whose restriction to K1 is not nilpotent, say a2 , and decompose K1 into a direct sum of A-submodules K2 ⊕ I2 as above. Obviously, the inductive procedure must terminate after finitely many steps and we get the desired decomposition. These considerations actually show slightly more. In order that condition (F) is satisfied it suffices that each member of the n-tuple a has finite ascent and finite descent. Corollary 5.4. If each member of the n-tuple a has finite ascent and finite descent, in particular, when the A-module M is both Artinian and Noetherian, then both ascent and descent of the n-tuple are finite and the conclusions of Theorem 5.2 are valid. Proof. The “Fitting” decomposition of M, given in condition (F) and guar(r) (r) anteed by our assumptions, decomposes Mn−1 = M ⊗A Dn−1 into (r) (r) (r) (r) Mn−1 = K ⊗A Dn−1 ⊕ I1 ⊗A Dn−1 ⊕ · · · ⊕ Ip ⊗A Dn−1 . (r) An element x ∈ ker(∂0r+1 ∂n−1 ) decomposes accordingly into x = u0 + u1 + (r) · · · + up . Since the differentials ∂n−1 and ∂0r+1 preserve the decomposition it follows that (r) (r) (r) (r) 0 = ∂0r+1 ∂n−1 x = ∂0r+1 ∂n−1 u0 + ∂0r+1 ∂n−1 u1 + · · · + ∂0r+1 ∂n−1 u0 . (r) For 0 ≤ i ≤ p the term ∂n−1 ui must therefore be in the kernel of each (r) aj restricted to K and Ii , respectively. This forces ∂n−1 ui to be zero for 1 ≤ i ≤ p, since then at least one of these restrictions is invertible. Hence (r) (r) ∂n−1 x = ∂n−1 u0 , and by the nilpotency of the restriction of each aj to K (r) (r) we see that ∂n−1 x = ∂n−1 u0 = 0 for r sufficiently large. Similarly, if y = (r) (r) ∂0 x ∈ im ∂0 , decompose x according to (F) into x = u0 +u1 +· · ·+up . For sufficiently large r we may assume u0 = 0 with no loss of generality. But then r+1 x ∈ im ∂n−1 , since the restriction ∂n−1 : ⊕pi=1 Ii ⊗A Dn−1 → ⊕pi=1 Ii ⊗A Dn is surjective. Examples 5.5. a) The converse of Corollary 5.4 is false. Here is an example. N , F ) be the vector space of all sequences in the field F . Consider Let V = F(N 110 L. GRUNENFELDER AND M. OMLADIČ the pair of commuting linear operators a = (a, b) on V defined by ( ( 0, for n = 1 0, for n = 1 (af )(n) = and (bf )(n) = . f (n), for n > 1 f (n + 1), for n > 1 Since a is idempotent, ker as = ker a = {f |f (n) = 0 for n > 1} and im as = im a = {f |f (1) = 0}, we see that the ascent and the descent of a are both equal to 1. On the other hand, ker br = {f |f (n) = 0 for n ≥ r + 2} and im br = im b = {f |f (1) = 0}, so that b has infinite ascent but descent 1. b) There is a pair of commuting endomorphisms on an infinite dimensional vector space M for which the ascent d0 = d0n−1,0 is infinite and the index d000 is finite. Construction. Let F be an arbitrary field and let M be the vector space of pairs (α, p), where α ∈ F and p = (pi (x)) is a sequence of polynomials with coefficients in F . If we define a : M → M by a(α, p1 (x), p2 (x), . . . ) = (0, xp1 (x), xp2 (x), . . . ) and b : M → M by b(α, p1 (x), p2 (x), . . . ) = (p1 (0), p2 (x), p3 (x), . . . ) then it is easy to verify that a and b are commuting endomorphisms of M. Moreover, we see that an (α, p1 (x), p2 (x), . . . ) = (0, xn p1 (x), xn p2 (x), . . . ) and hence ker an = {(α, 0)} for all n ≥ 1. Since ker a ⊂ ker b, we must necessarily have ∩i+j=n ker(ai bj ) = ker a for all n ≥ 1, showing that d000 = 1. In order to see that the ascent of this pair is infinite, choose for r ≥ 1 an element u = (α, p) ∈ M such that pr (0) 6= 0 and pk (x) = 0 for k > r. It then follows that 0 6= br u = (pr (0), 0, 0, . . . ) ∈ ker a. Let û = uê11 ⊗ ê21 ⊗ . . . ⊗ êr1 ∈ (r) (r) Mn−1 . Then br u = ∂n−1 ũ is a non-zero vector from ker ∂0r+1 = ker a. Hence, (r) (r) ker ∂n−1 6= ker(∂0r+1 ∂n−1 ) for every r ≥ 1. 6. Some remarks and comparisons. 6.1. If a is the ideal generated by a = (a1 , a2 , . . . , an ) and a(s) the ideal generated by as = (as1 , as2 , . . . , asn ) in A then clearly a(ns) ⊆ ans ⊆ a(s) ⊆ as , j (j) j (j) lim −→ j HomA (A/a , M) = lim −→ j HomA (A/a , M) and ∩j a M = ∩j a M. The local cohomology with coefficients in the A-module M of the ideal a is defined by ∗ s Ha∗ (M) = lim −→ s ExtA (A/a , M). An equivalent description in terms of Koszul complexes can be given as follows. For each s ≥ 1 the maps χ : K ∗ (M, as ) → K ∗ (M, as+1 ) and η : K ∗ (M, as+1 ) → K ∗ (M, as ), ASCENT AND DESCENT FOR FINITE SEQUENCES 111 defined by χ(mei1 ∧ ei2 ∧ . . . ∧ eil ) = ai1 ai2 . . . ail mei1 ∧ ei2 ∧ . . . ∧ eil and by η(mei1 ∧ ei2 ∧ . . . ∧ eil ) = ail+1 ail+2 . . . ain mei1 ∧ ei2 ∧ . . . ∧ eil , respectively, are cochain maps. It is known [BH, Theorem 3.5.6] that ∗ s ∗ ∗ s Ha∗ (M) = H ∗ (lim −→ s K (M, a )) = lim −→ s H (K (M, a )) and in particular, j (j) Ha0 (M) = lim −→ j HomA (A/a , M) = lim −→ j HomA (A/a , M), where the direct limit is taken along the maps χ. We may also take the inverse limit along the maps η and get a commutative diagram with exact rows 0 −−−→ ∩j a(j) M −−−→ y 0 −−−→ a(s) M n ∗ (j) M −−−→ lim ←− j H (K (M, a )) y −−−→ M −−−→ H n (K(M, a(s) )) −−−→ 0 for each s ≥ 1. 6.2. With the notation already used earlier we also have j (j) K=− lim → j HomA (A/a , M) = lim −→ j HomA (A/a , M), I = ∩j aj M = ∩j a(j) M and we get the obvious commutative diagram with exact rows 0 −−−→ K αy −−−→ M −−−→ M/K −−−→ 0 x β . 0 ←−−− M/I ←−−− M ←−−− I ←−−− 0 It is easy to see that: (1) K ∩ I = 0 iff α is injective iff β is injective; (2) K + I = M iff α is surjective iff β is surjective; (3) K ⊕ I = M iff α is bijective iff β is bijective. The ascending chain of submodules HomA (A/a, M) ⊆ . . . ⊆ HomA (A/aj , M) ⊆ HomA (A/aj+1 , M) ⊆ . . . of M becomes stationary if and only if the ascending chain HomA (A/a, M) ⊆ . . . ⊆ HomA (A/a(j) , M) ⊆ HomA (A/a(j+1) , M) ⊆ . . . becomes stationary. It follows from Proposition 4.1 that this happens at stage s when the ascent of a is finite and equal to s; in that case HomA (A/aj , M) = HomA (A/a(j) , M) 112 L. GRUNENFELDER AND M. OMLADIČ and HomA (A/aj , M) ∩ aj M = 0 for all j ≥ s. The descending chain of submodules M ⊇ aM . . . ⊇ aj M ⊇ aj+1 M ⊇ . . . becomes stationary if and only if the descending chain M ⊇ aM . . . ⊇ a(j) M ⊇ a(j+1) M ⊇ . . . becomes stationary. By Proposition 4.2 this happens at stage r when the descent of a is finite and equal to r; in that case aj M = a(j) M and HomA (A/aj , M) + aj M = M for all j ≥ r. In each case the two chains may not terminate at the same stage as the following example shows. If a1 and a2 are the linear operators defined by a1 (x1 , x2 , x3 , x4 ) = (x3 , x4 , 0, 0) and a2 (x1 , x2 , x3 , x4 ) = (x2 , 0, x4 , 0) on the vector space M = R4 , then a21 = 0 = a22 but a1 a2 (x1 , x2 , x3 , x4 ) = (x4 , 0, 0, 0), so that in each case the second chain terminates at stage two and the first at stage three. If both the ascent and the descent of a are finite then by Theorem 5.2 they are equal, K = HomA (A/as , M) = HomA (A/a(s) , M), I = as M = a(s) M and we have the Fitting decomposition HomA (A/as , M) ⊕ as M = M. However, Example 5.5. b) shows that the stationarity of the above chains of submodules of M alone will not guarantee a Fitting type decomposition of M. Therefore stationarity of these chains alone will not suffice to get a useful definition for finite ascent and finite descent. 7. Comparison maps. 7.1. Let a = (a1 , a2 , . . . , an ) be an n-tuple of commuting endomorphisms of the A-module M and let b = (b1 , b2 , . . . , bm ) be an m-tuple of commuting endomorphisms of the A-module N . A map of differential graded modules u : K(M, ∂a ) → K(N , ∂b ) must satisfy the equation ∂b u = u∂a , i.e. the diagram ∂ K(M, a) −−−→ K(M, a) u uy y ∂ K(N , b) −−−→ K(N , b) commutes. In particular, if u0 : M → N is an A-module map and U = [uij ] is an m×n-matrix of A-endomorphisms of N such that entries from different ASCENT AND DESCENT FOR FINITE SEQUENCES 113 rows commute, then a map of graded modules u = (u0 , U ) : K(M, a) → K(N , b) can be defined by u(vej1 ∧ ej2 ∧ . . . ∧ ejp ) X = ui1 j1 ui2 j2 . . . uip jp u0 (v)fi1 ∧ fi2 ∧ . . . ∧ fip i1 ,i2 ,... ,ip det([uik jl ]pk,l=1 )u0 (v)fi1 ∧ fi2 ∧ . . . ∧ fip , X = I∈J(p) where J(p) = {I = (i1 , i2 , . . . , ip )|1 ≤ i1 < i2 < . . . < ip ≤ n}. Using multi-indices this can be written as X uIJ u0 (v)fI , u(veJ ) = I∈J(p) where uIJ = det([uik jl ]pk,l=1 ). The following result is essentially [R, Lemma 2.1], where the condition that uij and bk commute for i 6= k should be added. Lemma 7.2. If the diagram ∂ M −−−→ K1 (M, a) u u0 y y 1 ∂ N −−−→ K1 (N , b) commutes, and if uij commutes with ukl and with bk for i 6= k, then u = (u0 , U ) : K(M, ∂a ) → K(N , ∂b ) is a cochain map. If, in addition, u0 is an isomorphism and the matrix [uij ] is invertible then u is an isomorphism of complexes. Proof. The commutativity of the diagram says that X X bi u0 (v)fi = ∂u0 (v) = u1 ∂(v) = uij u0 (aj v)fi i for every v ∈ M and hence bi u0 = i,j P i uij u0 aj for 1 ≤ i ≤ m. But then u∂(vej1 ∧ . . . ∧ ejp ) X = ui0 j0 ui1 ,j1 . . . uip ,jp u0 (aj0 v)fi0 ∧ fi1 ∧ . . . ∧ fip j0 ,i0 ,I = X i0 ,I bi0 ui1 ,j1 . . . uip ,jp u0 (v)fi0 ∧ fi1 ∧ . . . ∧ fip ,jp = ∂u(veJ ), 114 L. GRUNENFELDER AND M. OMLADIČ if uij and bk commute for i 6= k, so that the diagram ∂ Kp (M, a) −−−→ Kp+1 (M, a) up+1 up y y ∂ Kp (N , b) −−−→ Kp+1 (N , b) commutes for every p ≥ 0. In terms of multi-indices this reads X X u∂(veJ ) = u aj vej ∧ eJ = uij uIJ u0 (aj v)fi ∧ fI j = X j,i,I uIJ uij u0 (aj v)fi ∧ fI I,j,i = X bi uIJ u0 (v)fi ∧ fI = ∂u(veJ ). i,I If, in addition, u0 and [uij ] are invertible then a direct calculation shows that −1 −1 −1 : K(N , b) → K(M, a) is the inverse of u. (u−1 0 , [u0 uij u0 ] ) = u It is the following special case of the above situation that is of particular importance for us. Proposition 7.3. Let A and B be commutative R-subalgebras of EndA (M) containing the n-tuples a and b, respectively, such that bu0 = U u0 a for some n × n matrix U = [uij ] of A-endomorphisms and some A-endomorphism u0 of M. If uij commutes with ukl and with bk for k 6= i, then U induces a cochain map u = (u0 , U ) : K(M, ∂a ) → K(M, ∂b ). Moreover, if U is invertible and u0 is an automorphism then u is an isomorphism. 7.4. As the first of the following example shows, the conditions of Proposition 7.3 are not sufficient to guarantee that a and b have the same ascent or the same descent. In case M = R2 consider the matrices 0 1 0 1 −1 1 1 0 a= , b= , c= , u= 0 0 0 1 −1 1 1 1 in M2 (R). Then b = ua, a2 = 0, b2 = b and c = uau−1 . Thus, with (u0 , U ) = (id, u) the conditions of Proposition 7.3 are met, but the ascent and the descent of a are both equal to 2, while those of b are both equal to 1. On the other hand, the conditions are met again if we replace b by c = uau−1 and take (u0 , U ) = (u, id). As expected, a and c have of course the same ascent and the same descent. This, and considerations of how (u0 , U ) could induce a map of complexes from M(r) (a) to M(r) (b) for r ≥ 2, suggest that what is needed to guarantee ASCENT AND DESCENT FOR FINITE SEQUENCES 115 equal ascent and equal descent, aside from u0 being invertible, is that the entries of U commute pairwise and that they commute with the entries of b. It is obvious that ascent and descent do not depend on the particular subalgebra A of EndA (M) containing a. If u0 ∈ AutA (M) then bu0 = n n U u0 a is equivalent to b = U u0 au−1 0 . If, in addition, U : M → M is invertible then the entries of u0 au−1 0 commute with those of U and those of b. It therefore remains to compare ascent or descent of a, u0 au−1 0 and −1 b = U u0 au0 . Corollary 7.5. Let A and B be commutative R-subalgebras of EndA (M) containing the n-tuples a and b, respectively, such that bu0 = U u0 a for some U ∈ Mn (B) and some u0 ∈ AutA (M). Then (u0 , U ) induces a cochain map u(r) : M(r) (∂a ) → M(r) (∂b ) for each r ≥ 1. Moreover, if U ∈ GLn (B) then u(r) is an isomorphism for r ≥ 1 so that a and b have the same ascent and the same descent. Proof. Since U ∈ Mn (B), the conditions of Proposition 7.3 are obviously satisfied, and the assertion for r = 1 follows. Moreover, since the entries of U commute and they also commute with the entries of b it follows that X X bi u(veJ ) = bi uIJ u0 (v)fI = uIJ bi u0 (v)fI I = X I uIJ uij u0 (aj v)fI I,j = X uij uIJ u0 (aj v)fI = X uij u(aj veJ ), j j,I P which means that bi u = j uij uaj . Now proceed by induction on r. Suppose that u(r) : M(r) (a) → M(r) (b) has been defined such that ∂b u(r) = u(r) ∂a and that bi u(r) = P already (r) (r+1) : M(r+1) (a) → M(r+1) (b) by u(r+1) (x ⊗ j uij u aj . Then define u P eI ) = J uJI u(r) x ⊗ fJ . The commutativity conditions and the induction hypothesis then imply that X (r+1) (r+1) (r) ∂b u (x ⊗ eI ) = bj ∂b uJI u(r) x ⊗ (fj ∧ fJ ) j,J = X = X (r) bj uJI u(r) ∂a x ⊗ (fj ∧ fJ ) j,J (r) uji uJI u(r) ai ∂a x ⊗ (fj ∧ fJ ) i,j,J ! =u (r+1) X i (r) ai ∂a x ⊗ (ei ∧ eI ) 116 L. GRUNENFELDER AND M. OMLADIČ (r+1) = u(r+1) ∂a (x ⊗ eI ) and also bi u(r+1) (x ⊗ eI ) = X = X uJI bi u(r) x ⊗ fJ J J,i uJI uij u(r) (aj x) ⊗ fJ = X uij u(r+1) (aj x ⊗ eI ). j P (r+1) (r+1) (r+1) This means that ∂b u = u(r+1) ∂a and bi u(r+1) = j uij u(r+1) aj . If u0 ∈ AutA (M) and U ∈ GLn (B) then one shows directly that the inverse −1 −1 of u(r) is induced as above by (u−1 0 , [u0 uij u0 ] ) for each r ≥ 1. There is another, more conceptual way to show that u(r) is an isomorphism. It is easy to see that (u0 , id) induces an isomorphism M(r) (∂a ) ∼ = M(r) (∂u0 au−1 ). 0 (r) (∂ ) ∼ The fact that (id, U ) induces an isomorphism M(r) (∂ −1 ) = M b u0 au0 uses the assumption that the entries of u0 au−1 0 commute with those of U and those of b so that we may assume that all these entries are contained in B. The rest is then a consequence of the following general observation. If φ : X → X 0 and ψ : Y → Y 0 are maps of differential graded B-modules then φ ⊗B ψ : X ⊗B Y → X 0 ⊗B Y 0 is a map of bicomplexes and D(φ ⊗B ψ) : D(X ⊗B Y ) → (X 0 ⊗B Y 0 ) is a homomorphism of the associated diagonal complexes. Moreover, if φ and ψ are isomorphisms then so are φ ⊗B ψ and D(φ ⊗B ψ). 8. Invariance Properties. 8.1. In case u0 = id, the assumption that b = U a with U ∈ GLn (A) in 7.5 clearly implies that a and b generate the same ideal in A. The question now arises whether an n-tuple a and an m-tuple b, which generate the same ideal in A, also have the same ascent and the same descent. This is indeed the case, as we shall demonstrate using localization arguments together with the following basic result. Theorem. Let A be a commutative R-subalgebra of EndA (M) containing the n-tuple a and the m-tuple b. Suppose a and b generate the same ideal in A and that b = U a0 for some a0 ⊆ a and some U ∈ GLm (A). Then K(M, ∂a ) ∼ = K(M, ∂b )⊗K(Rn−m , 0), hence H(K(M, ∂a )) ∼ = H(K(M, ∂b )) n−m ⊗Λ(R ). Moreover, a and b have the same ascent and the same descent. Proof. In view of Corollary 7.5 we may assume that b = (a1 , a2 , . . . , am ), and that a and b generate the same ideal in A. Thus, if the n-tuple c = b∪0 is obtainedPfrom b by adjoining n − m zeros, then there is a system of equations ij=1 uij (a)aj = ci with uii = 1 for i = 1, 2, . . . , n . The matrix ASCENT AND DESCENT FOR FINITE SEQUENCES 117 U = [uij ] is in GLn (A), since det U = 1. So, by Corollary 7.5 we have K(M, ∂a ) ∼ = K(M, ∂c ) ∼ = K(M, ∂b ) ⊗ K(Rn−m , 0) as complexes and thus H(K(M, ∂a )) ∼ = H(K(M, ∂b )) ⊗ Λ(Rn−m ) as graded modules. Moreover, a and c have the same ascent and the same descent, and it then remains to show that the same is true for c and b. By induction we may assume that c is obtained from b by adjoining a single zero. For this purpose let D(b) = K(A, ∂b ) and E = K(R, 0), so that D(c) = K(A, ∂c ) = D(b)⊗E and moreover K(M, ∂b ) = M⊗A D(b), K(M, ∂c ) = M⊗A D(c) ∼ = M ⊗A D(b) ⊗ E. By the results of Section 4, we only have to analyse the diagram (r) Mm (c) (r) ∂m (c)y ∂ r+1 (c) (r) 0 Mm+1 (c) −− −−→ Mrm+1 (c) ⊗A D1 (c) to solve the ascent problem. First notice that D(c) = D(b) ⊗ E means in particular that Di (c) = (Di (b) ⊗ E0 ) ⊕ (Di−1 (b) ⊗ E1 ) ∼ = Di (b) ⊕ Di−1 (b) for 0 ≤ i ≤ m + 1, where of course Dj (b) = 0 for j < 0 and for j > m. Moreover, since the differential of E is zero, the diagram Di (c) ∂i (c)y ∼ = ∼ = ∼ = ∼ = −−−−→ (Di (b) ⊗ E0 ) ⊕ (Di−1 (b) ⊗ E1 ) −−−−→ Di (b) ⊕ Di−1 (b) ∂ (b)⊕∂ (b) i−1 y y i Di+1 (c) −−−−→ (Di+1 (b) ⊗ E0 ) ⊕ (Di (b) ⊗ E1 ) −−−−→ Di+1 (b) ⊕ Di (b) (r) commutes for every i, 0 ≤ i ≤ m. More generally, since Di (c) ∼ = (r) r ⊗A (Di (b) ⊕ Di−1 (b)), the r-fold tensor product, we see that ∂i (c) = ⊗rA (∂i (b) ⊕ ∂i−1 (b)). In particular for i = m and for i = 0, after expressing the right hand parts as direct sums, the diagrams (r) ∼ = (r) Dm (c) −−−→ Z ⊕ Dm−1 (b) , (r) (r) y(0,∂m−1 (b)) ∂m (c)y (r) ∼ = Dm+1 (c) −−−→ (r) Dm (b) (r) ∼ = (r) ∼ = D0 (c) −−−→ (r) ∂0 (c)y (r) D0 (b) (r) y(∂0 (b),0)t (r) D1 (c) −−−→ D1 (b) ⊕ U 118 L. GRUNENFELDER AND M. OMLADIČ (r) (r) commute. It follows that the differential ∂m (c) is related to ∂m−1 (b) by an isomorphism of the form ∼ = (r) (r) ∼ = (r) Mm (c) −−−→ M ⊗A Dm (c) −−−→ (M ⊗A Z) ⊕ Mm−1 (b) (r) (r) y(0,∂m−1 (b)) ∂m (c)y ∼ = (r) (r) ∼ = (r) Mm+1 (c) −−−→ M ⊗A Dm+1 (c) −−−→ Mm (b), while the relation between the differentials ∂0r+1 (c) and ∂0r+1 (b) is given by ∂ r+1 (c) (r) (r) 0 −− −−→ Mm+1 (c) ∼ =y Mm+1 (c) ⊗A D1 (c) ∼ y= (r) (r) M ⊗A Dm+1 (c) ∼ y= M ⊗A Dm+1 (c) ⊗A D1 (c) ∼ = y (∂ r+1 (b),0)t (r) (r) (r) −−0−−−−−→ (Mm (b) ⊗A D1 (b)) ⊕ (Mm (b) ⊗A D0 (b)). Mm (b) With the proper identifications we therefore get the matrix equation (r) (c) ∂0r+1 (c)∂m = (r) r+1 ∂0r+1 (b) (b)∂ (b) 0 ∂ (r) 0 m−1 . 0 ∂m−1 (b) = 0 0 0 (r) (r) (r) This shows that ker ∂m (c) = (M⊗A Z)⊕ker ∂m (b) and ker(∂0r+1 (c)∂m (c)) (r) = (M ⊗A Z) ⊕ ker(∂0r+1 (b)∂m (b)), and that the inclusion of the former into (r) (r) the latter is componentwise. Hence, ker ∂m (c) = ker(∂0r+1 (c)∂m (c)) if and (r) (r) only if ker ∂m (b) = ker(∂0r+1 (b)∂m (b)), which implies the invariance of ascent. To solve the descent problem we have, again by the results of Section 4, to analyse the diagram (r) r+1 ∂m (c) (r) M0 (c) ⊗A Dm (c) −−−−→ M0 (c) ⊗A Dm+1 (c) (r) y∂0 (c) (r) M1 (c) ⊗A Dm+1 (c). ASCENT AND DESCENT FOR FINITE SEQUENCES 119 r+1 (c) is related to ∂ r+1 (b) by the isomorphism Now, ∂m m−1 (r) M0 (c) ⊗A Dm (c) ∼ =y r+1 (c) ∂m −−−−→ (r) M ⊗A D(r) (c) ⊗A Dm+1 (c) ∼ y= M ⊗A D0 (c) ⊗A Dm (c) ∼ =y r+1 (b),0) (∂m−1 (r) (r) M0 (c) ⊗A Dm+1 (c) ∼ y= (M0 (b) ⊗A Dm−1 (b)) ⊕ V −−−−−−−→ (r) (r) M0 (b) ⊗A Dm (b) (r) with V = M(r) (b) ⊗A Dm (b), while ∂0 (c) and ∂0 (b) are related by (r) M0 (c) ⊗A Dm+1 (c) ∼ =y ∂ (r) (c) −−0−−→ (r) (r) M ⊗A D0 (c) ⊗A Dm+1 (c) ∼ =y (r) M0 (b) ⊗A Dm (b) (r) M1 (c) ⊗A Dm+1 (c) ∼ y= M ⊗A D1 (c) ⊗A Dm+1 (c) ∼ y= (∂ (r) (b),0)t (r) 0 −−− −−−−→ (M1 (b) ⊗A Dm (b)) ⊕ W if W = M ⊗A U ⊗A Dm (b). Considering the isomorphisms as identifications we have (r) (r) r+1 (r) (b) 0 ∂ (b)∂ ∂ (b) r+1 r+1 m−1 0 0 . ∂0 (c)∂m (c) = ∂m−1 (b) 0 = 0 0 0 (r) (r) (r) (r) r+1 (c)) = im(∂ (b)∂ r+1 (b)). Thus, im∂0 (c) = im∂0 (b) and im(∂0 (c)∂m m 0 (r) r+1 (c)) = im∂ (r) (c) if In particular, we may conclude now that im(∂0 (c)∂m 0 (r) (r) r+1 (b)) = im∂0 (b), establishing the invariance of and only if im(∂0 (b)∂m−1 descent. Corollary 8.2. Let R be a field and let L be a finite dimensional subspace of commuting elements in EndA (M). Any two finite spanning subsets of L have the same ascent and the same descent. Proof. If a and b are two finite spanning subsets of L then they generate the same commutative subalgebra A and the same ideal in it. Moreover, they each contain a basis a0 and b0 of L, so that b0 = U a0 for some U ∈ GLn (R), where n = dimR L. The assertion now follows from Theorem 8.1. 8.3. Additional conditions on the commuting n-tuple a or the commutative subalgebra A of EndA (M) it generates may give tighter invariance results. Such a situation occurs when A is a commutative local algebra; this happens for example when R is a field and a consists of nilpotent endomorphisms. 120 L. GRUNENFELDER AND M. OMLADIČ Theorem. Suppose A is a commutative local subalgebra of EndA (M) with maximal ideal M and residue class field F and let I be a finitely generated ideal of A. Then any two finite generating subsets of I have the same ascent and the same descent. Proof. The n-tuple a = (a1 , a2 , . . . , an ) ⊂ I forms a generating subset for the ideal I of the commutative local algebra A if and only if the residue classes ã = (ã1 , ã2 , . . . , ãn ) span I/M I as a vector space over the field F = A/M , i.e. if and only if ã contains an F -basis of I/M I. This follows from Nakayama’s Lemma [A, S]. Thus, if this is the case, then a contains a subset a0 which generates the ideal I and such that ã0 is an F -basis of I/M I. By Theorem 8.1 we see that a and b = a0 have the same ascent and the same descent. It remains to show that the same is true for m-tuples a and b for which ã and b̃ are F -bases of I/M I. But in this case b = U a ^ for some matrix U ∈ Mm (A) with Ũ ∈ GLm (F ). Then det U = det Ũ 6= 0, so that det U ∈ / M . Hence det U is invertible and U ∈ GLm (A). Again by Theorem 8.1 we conclude that a and b have the same ascent and the same descent. 8.4. We are now in a position to prove a general invariance result for any finitely generated ideal in a commutative subalgebra A of EndA (M). Theorem. Let A be a commutative R-subalgebra of EndA (M) and let J be a finitely generated ideal of A. Then any two finite generating subsets of J have the same ascent and the same descent. Proof. Suppose that J can be generated by the n-tuple a = (a1 , a2 , . . . , an ) as an ideal of A. For any prime ideal P of A let AP and MP = AP ⊗A M denote the localizations of A and M at P . The ideal I = JAP of the local ring AP is then generated by a as well. We shall see that dj (a, M) = sup{dj (a, MP )} P for j = 0, 1. Since the localization functor ( )P : ModA → ModAP is exact [A, Proposition 3.9], [S], we see that K (r) (M, ∂a )P ∼ = K (r) (MP , ∂a ), (r) (r) (r) (r) ∼ ∼ H (M, ∂a )P = H (MP , ∂a ), L (M, ∂a )P = L (MP , ∂a ) and R(r) (M, ∂a )P ∼ = R(r) (MP , ∂a ) for all r ≥ 1. Thus, by Propostions 4.1 and 4.2, we see that dj (a, M) ≥ dj (a, MP ) for every prime ideal P of A. But, by [A, Proposition 3.9], [S], a homomorphism of A-modules f : X → Y is an isomorphism if and only if for each prime ideal P of A the induced homomorphism of AP -modules fP : XP → YP is an isomorphism. By the definition of ascent and descent in 4.3 we conclude that dj (a, M) = dj (a, MP ) ASCENT AND DESCENT FOR FINITE SEQUENCES 121 for some prime ideal P of A. If b is another generating subset of the ideal J then invoke Theorem 8.3 to see that dj (a, M) = sup{dj (a, MP )} = sup{dj (b, MP )} = dj (b, M) P P which proves our assertion. References [A] M. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Benjamin, New York, 1972. [B] D. Benson, Representations and Cohomology, Cambridge University Press, Cambridge, 1991. [BT] T. Bhattacharyya, On tuples of commuting compact operators, Publ. RIMS. Kyoto Univ., 32 (1996), 785-795. [BH] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, Cambridge, 1993. [C] R.B. Curto, Spectral Theory of Elementary Operators, in ‘Elementary operators and applications’, ed. M. Mathieu, Proceedings of International Workshop, World Scientific, Singapore, (1992), 3-52. [P] R.S. Pierce, Associative Algebras, Springer, New York, 1982. [R] B.P. Rynne, Multiparameter spectral theory and Taylor’s joint spectrum in Hilbert space, Proc. Edin. Math. Soc., 31 (1988), 122-144. [S] J.P. Serre, Algèbre Locale et Multiplicités, Springer Lecture Notes, 11 (1965). [TAE] A.E. Taylor, Introduction to Functional Analysis, Wiley and Sons, New York, 1958. [T] J.L. Taylor, A joint spectrum for several commuting operators, J. Funct. Anal., 6 (1970), 172-191. Received February 20, 1998 and revised August 18, 1998. This research supported in part by NSERC of Canada and the Ministry of Science and Technology of Slovenia. Department of Mathematics, Statistics and Computing Science Dalhousie University Halifax, N.S. B3H 3J5 Canada E-mail address: [email protected] Department of Mathematics University of Ljubljana 61000 Ljubljana Slovenia PACIFIC JOURNAL OF MATHEMATICS Vol. 191, No. 1, 1999 TOPOLOGY VERSUS CHERN NUMBERS FOR COMPLEX 3-FOLDS Claude LeBrun We show by example that the Chern numbers c31 and c1 c2 of a complex 3-fold are not determined by the topology of the underlying smooth compact 6-manifold. In fact, we observe that infinitely many different values of a Chern number can be achieved by (integrable) complex structures on a fixed 6manifold. 1. Introduction. Suppose that X is a smooth compact oriented 6-manifold. Recall that an almost-complex structure on X means an endomorphism J : T X → T X of the tangent bundle of X with J 2 = −1 which determines the given orientation of X. Such a structure makes T X into a complex vector bundle, so that one can speak of the Chern classes cj ∈ H 2j (X, Z) of (X, J), and therefore of the Chern numbers Z 3 c1 = c31 ZX c 1 c2 = c1 c2 ZX c3 = c3 X of the almost-complex manifold (X, J). The only obstruction [10] to the existence of an almost-complex structure J on X is that X be spinc . This happens precisely when the second Stiefel-Whitney class w2 (X) ∈ H 2 (X, Z2 ) can be written as the mod-2 reduction of an element of H 2 (X, Z), in which case each preimage of w2 in H 2 (X, Z) can be realized as c1 for some almostcomplex structure J. It follows that the Chern numbers c31 and c1 c2 of the almost-complex (X, J) are certainly not topological invariants of the 6-manifold X. For example, if X = CP3 , every integer of the form 8j can be realized as c1 c2 , and every integer of the form 8j 3 can be realized as c31 for some almost-complex structure J on CP3 . On the other hand, c3 is the Euler class of T X, so that c3 = χ(X) is actually a homotopy invariant of X. 123 124 CLAUDE LEBRUN In this note, we will observe that the above situation persists even if one demands that the almost-complex structures under consideration be integrable. Recall that an almost-complex structure J on X is called a complex structure if it is integrable, in the sense of being locally isomorphic to the standard, constant-coefficient structure on R6 = C3 . The question of whether the Chern numbers c31 and c1 c2 of a complex 3-fold might actually be topological invariants of the underlying 6-manifold was raised, for example, in an interesting survey article by Okonek and van de Ven [7, p. 317]. Our principal results are as follows: Theorem A. There is a compact simply connected 6-manifold X which admits a sequence Jm , m ∈ Z+ , of (integrable) complex structures with c1 c2 (X, Jm ) = 48m. Indeed, there are infinitely many homeotypes of X with this property. Theorem B. Let (m, n) be any pair of integers. Then for any integer ñ n, there is a complex projective 3-fold (X, J) with Chern numbers c1 c2 = 24m, c31 = 8n, which admits a second complex structure J˜ with c1 c2 = 48m, c31 = 8ñ. 2. Infinitely Many Complex Structures. The fact that the Chern classes of a complex 3-fold are not determined by the topology of the underlying 6-manifold was observed long ago by Calabi [2]. While his examples all have vanishing Chern numbers, they nonetheless contain the seeds of a natural class of examples which lead to Theorem A: Theorem 1. For each positive integer m, the 6-manifold X = K3 × S 2 admits a complex structure Jm with c1 c2 (X, Jm ) = 48m and c31 (X, Jm ) = 0. Proof. Let M denote the underlying oriented 4-manifold of the K3 surface, and let g be any hyper-Kähler metric on M ; such metrics exist by Yau’s solution of the Calabi conjecture [11]. Let Z be the twistor space [1, 8] of (M, g), and let $ : Z → CP1 be the holomorphic projection induced by the hyper-Kähler structure. Differentiably, $ is the trivial fiber bundle with fiber M , so that Z may be thought of as X = M × S 2 equipped with a complex structure. Now let fm : CP1 → CP1 be a holomorphic map of degree m − 1; for example, we may take fm ([u, v]) = [um−1 , v m−1 ], where m is any positive COMPLEX 3-FOLDS 125 ∗ $ of K3’s over CP integer. We may then define a holomorphic family fm 1 by pulling back the family $ via fm : In other words, ∗Z fm ∗Z fm ∗$ ↓ fm −→ CP1 −→ $ fm Z ↓ CP1 . is the inverse image, via 1 × $ : CP1 × Z−→ CP1 × CP1 , ∗ $, and f ∗ Z of the graph of fm . Since $ is differentiably trivial, so is fm m may therefore be viewed as X = K3×S 2 equipped with a complex structure Jm . Now if π : Z → M is the (non-holomorphic) twistor projection, an explicit ∗ Z is similarly trivialized by diffeomorphism Z → X is given by π ×$, and fm ∗ $×f ∗ π. Let L ⊂ T f ∗ Z be the kernel of the derivative of the pulled-back fm m m ∗ π. Then L is J invariant, despite the fact that π is not twistor projection fm m holomorphic, and so may be viewed as a complex line-bundle. Moreover, L may be identified with the pull-back of the (holomorphic) tangent bundle of ∗ $, so that c2 (L) = 0, and hence p (L) = 0. If, on the other hand, CP1 via fm 1 1 ∗ $, then the underlying we use H to denote the kernel of the derivative of fm ∗ ∗ ∗ real bundle of H is (fm π) T M , and so p1 (H) = fm [p1 (M )] = −48F , where, ∗ π. It follows that by Poincaré duality, F is represented by a fiber S 2 of fm ∗ p1 (fm Z) = p1 (L) + p1 (H) = −48F . On the other hand, any K3 has trivial canonical line bundle, and the fibers of π are CP1 ’s with normal bundle O(1) ⊕ O(1), so c1 (H) = c1 (L) when m = 2. For general m, it follows that c1 (H) = (m − 1)c1 (L), and hence that c1 = mc1 (L). We therefore have c21 = c1 c2 = and c31 = 0. m2 c21 (L) = c31 − c1 p1 2 0, so that Z = 24 mc1 (L) = 48m, F When m = 1, the above complex structure is simply an arbitrary product complex structure on K3 × CP1 , and so is of Kähler type; indeed, we may even arrange for it to be projective-algebraic if we like. On the other hand, the m = 2 complex structure is that of a twistor space, and so is never of Kähler type by Hitchin’s classification of Kählerian twistor spaces [4]. For large values of m, one can prove something even stronger: Jm isn’t even homotopic to a complex structure of Kähler type. This is because the Todd genus c1 c 2 1 − h1 (O) + h2 (O) − h3 (O) = χ(O) = = 2m, 24 126 CLAUDE LEBRUN so that h2 (O) will eventually exceed b2 (X), in violation of the Hodge decomposition. In the next section, we will see that this phenomenon is actually quite typical. In order to show that there is more than one 6-manifold for which infinitely many different values of a Chern number are achieved by (integrable) complex structures, we may now invoke the standard process of blowing up. The following facts about blow-up 3-folds are left as exercises for the reader. ˆ Proposition 2. Let (X, J) be any compact complex 3-fold, and let (X̂, J) be obtained from (X, J) by blowing up a point. Then X̂ is diffeomorphic to the connected sum X#CP3 , and if X is spin, so is X̂. Moreover, the Chern numbers of the blow-up are related to those of the original 3-fold by ˆ = c31 (X, J) + 8 c31 (X̂, J) ˆ = c1 c2 (X, J) c1 c2 (X̂, J) ˆ = c3 (X, J) + 2. c3 (X̂, J) Iterated blow-ups of the previous examples thus prove the following precise form of Theorem A: Corollary 3. For each integer n ≥ 0, the 6-dimensional spin manifold X = (K3 × S 2 )#nCP3 admits a sequence Jm of complex structures with c1 c2 (X, Jm ) = 48m c31 (X, Jm ) = 8n. 3. Kähler Type. We saw in Theorem 1 that it is possible to find 6-manifolds with sequences of complex structures for which a Chern number takes on infinitely many different values. On the other hand, if one requires that the complex structures in question be of Kähler type, one arrives at essentially the opposite conclusion. This is illustrated by our next result. Theorem 4. Let X be the underlying compact oriented 6-manifold of any Kählerian 3-fold. Then there exist infinitely many homotopy classes of almost-complex structures on X which cannot be represented by complex structures of Kähler type. Proof. By assumption, there is a Kähler class [ω] ∈ H 2 (X, R) with [ω]3 6= 0. Now H 2 (X, Q) ⊂ H 2 (X, R) is dense, and the cup form is continuous, so approximating [ω] with rational classes will produce classes α0 ∈ H 2 (X, Q) with α03 > 0. Multiplying by a suitable positive integer k to clear denominators, we may thus obtain a class kα0 which is the image of an integer COMPLEX 3-FOLDS 127 class α ∈ H 2 (X, Z) in rational cohomology. Now let β ∈ H 2 (X, Z) be the first Chern class of the given complex structure on X. Then 2nα + β is an integer lift of w2 , and so [10] can be realized as c1 for some homotopy class [Jn ] of almost-complex structures. Now if Jn ∈ [Jn ] is integrable, the Todd genus of (X, Jn ) is c1 · c2 (2nα + β)3 − (2nα + β) · p1 = 24 48 which is cubic in n, with the coefficient of n3 non-vanishing. It P therefore P follows that there is an integer n0 such that | k (−1)k h0,k | > j bj (X) whenever |n| > n0 . For n in this range, the Hodge theorem must therefore fail, and so an integrable Jn could not possibly be of Kähler type. χ(O) = The reader should note that we have only used two mild consequences of the Kähler condition: the degeneration of the Fröhlicher spectral sequence, and the non-triviality of the cup form on H 2 . The same argument would thus apply if one instead wished to consider, say, complex structures of Moishezon type. 4. Independence of Chern Numbers. So far, we have seen that the Chern number c1 c2 of a complex 3-fold is not an invariant of the underlying 6-manifold. We will now see see that the same is true of c31 . To this end, let N be any smooth, compact oriented 4-manifold. By [9], the connected sum M = N #kCP2 admits anti-self-dual metrics g provided that k is sufficiently large. The twistor space of such an anti-self-dual metric is a complex 3-fold (Z, J2 ), the underlying 6-manifold Z of which is formally the fiber-wise projectivization P(S+ ) of the bundle of positive spinors on M . This description may seem a bit paradoxical, insofar as we are interested in choices of M which definitely are not spin, but it may be made quite concrete by choosing a spinc structure on M . This then gives rise to a well defined “twisted spinor” bundle V+ which formally satisfies V+ = S+ ⊗ L1/2 for a line bundle L with c1 (L) ∼ = w2 (M ) mod 2. This done, we then have a canonical identification of Z with the total space of the CP1 -bundle P(V+ ). The naturally defined complex structure J2 then makes each fiber of the projection π : Z → M into a holomorphically embedded CP1 with normal bundle O(1) ⊕ O(1). For more details, see [1, 4, 8]. Now let us now specialize to the case in which N is a complex surface, and notice that M = N #kCP2 may then be thought of as an iterated blow-up of N , and so, in particular, carries a complex structure. This complex structure induces a spinc structure on M such that, for any metric g, the associated 128 CLAUDE LEBRUN twisted spin bundle V+ is smoothly bundle-isomorphic to the holomorphic vector O ⊕ K −1 , where K is the canonical line bundle of M . Indeed, for a Hermitian metric on M , there is even a canonical isomorphism V+ ∼ =O⊕ K −1 ; and, up to abstract the bundle equivalence, the twisted spinor bundle V+ is metric-independent once a spinc structure is specified. In this way, the twistor space Z of an anti-self-dual metric g on M is diffeomorphic to the complex manifold P(O⊕K −1 ), and so carries a second complex structure J1 . Notice that we do not need to assume any compatibility between the metric g and the complex structure of M . Also notice that (Z, J1 ) is projective algebraic (respectively, Kählerian) if M is. Let us now calculate the Chern numbers of (Z, J1 ). To do this, first notice that Z = P(O ⊕ K −1 ) carries two canonical hypersurfaces, Σ and Σ, corresponding to the factors of the direct sum O ⊕ K −1 . These are both copies of the complex surface M , but their normal bundles are respectively K −1 and K. Moreover, the divisor Σ + Σ precisely represents the vertical line bundle L of Z. We thus have c1 (Z, J1 ) = Σ + Σ + π ∗ c1 (M ) = 2Σ. Hence c31 (Z, J1 ) = (2Σ)3 = 8(2χ + 3τ ) and c1 c2 (Z, J1 ) = 2(c21 + c2 )(M ) = 6(χ + τ ), where χ and τ are the Euler characteristic and signature of M , respectively. On the other hand, there is a fiber-wise antipodal map which acts antiholomorphically on (Z, J2 ), so c1 (J2 ) is Poincaré dual to an element of H4 (Z) which is invariant under this antipodal map. We also know that the integral of c1 (J2 ) on a fiber is 4. It follows that c1 (Z, J2 ) = 2Σ + 2Σ. Since the restrictions of J1 and J2 to a tubular neighborhood of Σ are also homotopic, we therefore deduce the formulæ c31 (Z, J2 ) = 16(2χ + 3τ ) c1 c2 (Z, J2 ) = 12(χ + τ ) derived (with opposite orientation conventions) by Hitchin [4] in greater generality. For us, the point is that the invariants c31 and c1 c2 of (Z, J2 ) are precisely double the corresponding Chern numbers of (Z, J1 ). We are now in a position to prove a more precise version of Theorem B. COMPLEX 3-FOLDS 129 Theorem 5. Let (m, n) be any pair of integers. Then for any integer ñ n, there is a spin, complex projective 3-fold (X, J) with Chern numbers c1 c2 = 24m, c31 = 8n, which admits a second complex structure J˜ with c1 c2 = 48m, c31 = 8ñ. If m > 0, moreover, we may even arrange for X to be simply connected. Proof. For each integer m, we begin by choosing a complex algebraic surface Nm with Todd genus (χ + τ )/4 = m. For example, if m ≤ 1, let us take N to be C × CP1 , where C is a Riemann surface of genus 1 − m. On the other hand, if m > 1, we may let N be the minimal resolution of (E×C)/Z2 , where E is an elliptic curve, C is a hyperelliptic curve of genus m − 1, and Z2 acts simultaneously on both factors by the Weierstrass involution. Notice that our choice of Nm is simply connected when m > 0, and that, incidentally, this is the best one can do in principal. Now, for each m, let k0 (m) be chosen so that Nm #kCP2 admits anti-selfdual metrics for each k ≥ k0 (m). By Taubes’ theorem [9], such an integer k0 (m) exists. Moreover, with the above choices, we may even take k0 (m) = 0 for m < 0, k0 (0) = 6, k0 (1) = 13, and k0 (2) = 3 [5, 6, 3]. Now let (m, n) be any pair of integers, and let ñ be any integer such that ñ ≤ min(n − k0 (m) + c21 (Nm ), 2n). We may then define integers k ≥ k0 (m) and ` ≥ 0 by k = n − ñ + c21 (Nm ), ` = 2n − ñ. Let Z(k, m) be the twistor space of an anti-self-dual metric on M = Nm #kCP2 , and let X(k, `, m) = Z(k, m)#`CP3 . Notice that X is a spin manifold. Moreover, it comes equipped with two different complex structures. First, because M = Nm #kCP2 is a projective algebraic surface, Z carries a projective algebraic complex structure J1 , and X may then be identified with the blow-up of (Z, J1 ) at ` points. Let us denote this complex structure on X by J. Using Proposition 2 and the above computations, we therefore have c31 (X, J) = 8(2χ + 3τ )(M ) + 8` = 8n c1 c2 (X, J) = 6(χ + τ )(M ) = 24m. On the other hand, each Z also admits its twistor complex structure J2 , and we may instead choose to think of X as the blow-up of this twistor space 130 CLAUDE LEBRUN ˜ at ` points. Let us denote the corresponding complex structure on X by J. Thus ˜ = 16(2χ + 3τ )(M ) + 8` = 8ñ, c31 (X, J) ˜ = 12(χ + τ )(M ) = 48m, c1 c2 (X, J) as claimed. Since the Todd genus of any complex 3-fold is given by c1 c2 /24, this result realizes all possible values of c1 c2 . The divisibility of c31 by 8 is also necessary for X to be spin, so the result is also essentially optimal in this respect. On the other hand, we have chosen to ignore c3 , which is determined by (m, n, ñ) in these examples. The abundance of rational curves also forces all our 3-folds all have Kodaira dimension −∞. And finally, most of our manifolds are in no sense minimal. It would obviously be of great interest to produce new examples which overcome these limitations. Acknowledgments. The author would like to thank Ron Stern for drawing his attention to the problem and pointing out several important references. He would also like to the thank the Mathematics Department of Harvard University for its hospitality during the inception of this work. References [1] M. Atiyah, N. Hitchin and I. Singer, Self-duality in four dimensional Riemannian geometry, Proc. R. Soc. Lond., A 362 (1978), 425-461. [2] E. Calabi, Construction and properties of some 6-dimensional almost complex manifolds, Trans. Amer. Math. Soc., 87 (1958), 407-438. [3] S. Donaldson and R. Friedman, Connected sums of self-dual manifolds and deformations of singular spaces, Nonlinearity, 2 (1989), 197-239. [4] N.J. Hitchin, Kählerian twistor spaces, Proc. Lond. Math. Soc., 43 (1981), 133-150. [5] C. LeBrun, Scalar-flat Kähler metrics on blown-up ruled surfaces, J. Reine Angew. Math., 420 (1991), 161-177. [6] C. LeBrun and M.A. Singer, A Kummer-type construction of self-dual 4-manifolds, Math. Ann., 300 (1994), 165-180. [7] C. Okonek and A. Van de Ven, Cubic forms and complex 3-folds, Enseign. Math., 41(2) (1995), no. 3-4, 297-333. [8] R. Penrose, Non-linear gravitons and curved twistor theory, Gen. Rel. Grav., 7 (1976), 31-52. [9] C.H. Taubes, The existence of anti-self-dual metrics, J. Diff. Geom., 36 (1992), 163253. [10] C.T.C. Wall, Classification problems in differential topology. V. On certain 6manifolds, Invent. Math., 1 (1966), 355-374; 2 (1966), 306. COMPLEX 3-FOLDS 131 [11] S.-T. Yau, On the Ricci-curvature of a complex Kähler manifold and the complex Monge-Ampère equations, Comment. Pure Appl. Math., 31 (1978), 339-411. Received February 19, 1998. This author was supported in part by NSF grant DMS9505744. SUNY Stony Brook Stony Brook, NY 11794 E-mail address: [email protected] PACIFIC JOURNAL OF MATHEMATICS Vol. 191, No. 1, 1999 COHOMOLOGY OF COMPLETE INTERSECTIONS IN TORIC VARIETIES Anvar R. Mavlyutov We explicitly describe cohomology of complete intersections in compact simplicial toric varieties. In this paper we will study intersections of hypersurfaces in compact simplicial toric varieties PΣ . The main purpose is to relate naturally the Hodge structure of a complete intersection Xf1 ∩ . . . ∩ Xfs in PΣ to a graded ring. Originally this idea appears in [Gr], [St], [Dol], [PS]. The case of a hypersurface in a toric variety has been treated in [BC]. Also the Hodge structure of complete intersections in a projective space was described in [Te], [Ko], [L], [Di], [Na]. The common approach was to reduce studying of the Hodge structure on a complete intersection to studying of the Hodge structure on a hypersurface in a higher dimensional projective variety. This is the idea of a “Cayley trick”. About a Cayley trick in the toric context see [GKZ], [DK], [BB]. A special case of a complete intersection (when it is empty) in a complete simplicial toric variety was elaborated in [CCD]. The basic references on toric varieties are [F1], [O], [Da], [C]. The paper is organized as follows: Section 1 establishes notation and studies cohomology of subvarieties in a complete simplicial toric variety. In Section 2 we describe a Cayley trick for toric varieties. In Section 3 we prove the main result where we relate the Hodge components H d−s−p,p (Xf1 ∩ · · · ∩ Xfs ) in the middle cohomology group to homogeneous components of a graded ring. Section 4 treats a special case of complete intersections: a nondegenerate intersection. Acknowledgment. I would like to thank D. Cox for his advice and useful comments. 1. Quasi-smooth intersections. We first fix some notation. Let M be a lattice of rank d, N = Hom(M, Z) the dual lattice; MR (resp. NR ) denotes the R-scalar extension of M (resp. of N ). Let Σ be a rational simplicial complete d-dimensional fan in NR [BC], PΣ a complete simplicial toric variety associated with this fan. Such a toric variety can be described as a geometric quotient [C]. Let S(Σ) = C[x1 , . . . , xn ] be the polynomial ring over C with variables x1 , . . . , xn 133 134 ANVAR R. MAVLYUTOV corresponding to the integral Q generators e1 , . . . , en of the 1-dimensional cones of Σ. For σ ∈ Σ let x̂σ = ei ∈σ / xi , and let B(Σ) = hx̂σ : σ ∈ Σi ⊂ S be the ideal generated by the x̂σ ’s. This ideal gives the variety Z(Σ) = V(B(Σ)) ⊂ An . The toric variety P = PΣ will be a geometric quotient of U (Σ) := An \ Z(Σ) by the group D := HomZ (Ad−1 (P), C∗ ), where Ad−1 (P) is the Chow group of Weil divisors modulo rational equivalence. Each variable xi in the coordinate ring S(Σ) corresponds to a torusinvariant irreducible divisor Q Di of P. As in [C], P we grade S = S(Σ) by assigning to a monomial ni=1 xai i its degree [ ni=1 ai Di ] ∈ Ad−1 (P). A polynomial f in the graded piece Sα corresponding to α ∈ Ad−1 (P) is said to be D-homogeneous of degree α. Let f1 , . . . , fs be D-homogeneous polynomials. They define a zero set V(f1 , . . . , fs ) ⊂ An , moreover V(f1 , . . . , fs ) ∩ U (Σ) is stable under the action of D and hence descends to a closed subset X ⊂ P, because P is a geometric quotient. Definition 1.1. We say that X is a quasi-smooth intersection if V(f1 , . . . , fs ) ∩ U (Σ) is either empty or a smooth subvariety of codimension s in U (Σ). Remark 1.2. This notion generalizes a nonsingular complete intersection in a projective space. Notice that since the (n − d)-dimensional group D has only zero dimensional stabilizers [BC], X is of pure dimension d − s or empty. We can now relate this notion to a V-submanifold (see Definition 3.2 in [BC]). Proposition 1.3. If X ⊂ P is a closed subset of codimension s defined by D-homogeneous polynomials f1 , . . . , fs , then X is a quasi-smooth intersection if and only if X is a V-submanifold of P. The proof of this is very similar to the proof of the Proposition 3.5 in [BC]. The next result is a Lefschetz-type theorem. Proposition 1.4. Let X ⊂ P be a closed subset, defined by D-homogeneous polynomials f1 , . . . , fs , in a complete simplicial toric variety P. If f1 , . . . , fs ∈ B(Σ), then the natural map i∗ : H i (P) → H i (X) is an isomorphism for i < d − s and an injection for i = d − s. In particular, this is valid if X is an intersection of ample hypersurfaces. Proof. We can present X = Xf1 ∩ . . . ∩ Xfs , where Xfi ⊂ P is a hypersurface defined by fi . As it was shown in the proof of the Proposition 10.8 [BC], if f ∈ B(Σ) then P \ Xf = (An \ V(f ))/D(Σ) is affine, hence H i (P \ Xf ) = 0 for i > d. We will prove by induction on s that H i (P \ (Xf1 ∩ . . . ∩ Xfs )) = 0 for i > d + s − 1. Consider the Mayer-Vietoris sequence · · · → H i (U ∩V ) → H i+1 (U ∪V ) → H i+1 (U )⊕H i+1 (V ) → H i+1 (U ∩V ) → · · · COHOMOLOGY OF COMPLETE INTERSECTIONS IN TORIC VARIETIES 135 with U = P\(Xf1 ∩. . .∩Xfs−1 ), V = P\Xfs . Notice that U ∪V = P\(Xf1 ∩ . . .∩Xfs ) and U ∩V = ∪s−1 i=1 P \(Xfi ∪Xfs ) = P \ (Xf1 ·fs ∩. . .∩Xfs−1 ·fs ). So, using the induction and the above sequence, we obtain that H i (P\X) = 0 for i > d + s − 1. As a consequence of this, X is nonempty unless s > d because the dimension h2d (P) = 1. Since P \ X is a V-manifold, Poincaré duality implies that Hci (P \ X) = 0 for i ≤ d − s. Now the desired result follows from the long exact sequence of the cohomology with compact supports (X and P are compact): · · · → Hci (P \ X) → Hci (P) → Hci (X) → Hci+1 (P \ X) → Hci+1 (P) → · · · . If X is an intersection of ample hypersurfaces defined by f1 , . . . , fs , then Lemma 9.15 [BC] gives us that f1 , . . . , fs belong to B(Σ). Corollary 1.5. A quasi-smooth intersection X = Xf1 ∩ . . . ∩ Xfs , defined by f1 , . . . , fs ∈ B(Σ), has pure dimension d − s. Since the dimension of H 0 (X, C) is the number of connected components of X, we obtain another important result. Corollary 1.6. An intersection Xf1 ∩ . . . ∩ Xfs , defined by f1 , . . . , fs ∈ B(Σ), in a complete simplicial toric variety PΣ is connected provided s < dim PΣ . Remark 1.7. If the polynomials f1 , . . . , fs have ample degrees, then this corollary follows from a more general statement in [FL1] (see also [FL2] and [FH] for connectedness theorems). 2. “Cayley trick”. We will explore a Cayley trick to reduce studying of the cohomology of quasi-smooth intersections to results already known for hypersurfaces. Let L1 , . . . , Ls be line bundles on a complete d-dimensional toric variety P = PΣ , and let π : P(E) → P be the projective space bundle associated to the vector bundle E = L1 ⊕ · · · ⊕ Ls . Then the Ps−1 -bundle P(E) is a toric variety. The fan corresponding to it can be described as follows [O, p. 58]. Suppose that support functions h1 , . . . , hs give rise to the isomorphism classes of line bundles [L1 ], . . . , [Ls ] ∈ Pic(P), respectively. Introduce a Z-module N 0 with a Z-basis {n2 , . . . , ns } and let Ñ := N ⊕ N 0 and n1 := −n2 − · · · − ns . Denote by σ̃ the image of each P σ ∈ Σ under the R-linear map NR → ÑR which sends y ∈ NR to y − sj=1 hj (y)nj . On the 0 generated by n , . . . , n , n other hand, let σi0 be the cone in NR 1 i i+1 , . . . , ns 0 0 and let Σ be the fan in NR consisting of the faces of σ10 , . . . , σs0 . Then P(E) corresponds to the fan Σ̃ := {σ̃ + σ 0 : σ ∈ Σ, σ 0 ∈ Σ0 }. From this description it is easy to see that if Σ is a complete simplicial fan then P(L1 ⊕ · · · ⊕ Ls ) is a complete simplicial toric variety. We see that the integral generators of 136 ANVAR R. MAVLYUTOV the 1-dimensional cones in Σ̃ are given by X hj (ei )nj , ẽi = ei − i = 1, . . . n, 1≤j≤s ñ1 = −n2 − · · · − ns , ñj = nj , j = 2, . . . , s, where e1 , . . . , en are the integral generators of the 1-dimensional cones in Σ. The homogeneous coordinate ring of P(E) is the polynomial ring R = C[x1 , . . . , xn , y1 , . . . , ys ], where xi corresponds to ẽi and yj corresponds to ñj . This ring has a grading by the Chow group Ad+s−2 (P(E)). Since P is a normal variety, there is an embedding of the Picard group Pic(P) ,→ Ad−1 (P). We want to show that if some polynomials fj ∈ S(Σ) = C[x1 , . . . , xn ] have the property deg(fj ) = [Lj ] ∈ Pic(P), then the polynomials yj fj all have the same degree in R. This will allow us Pto consider a hypersurface defined by the homogeneous polynomial F = sj=1 yj fj . Lemma 2.1. Let f1 , . . . , fs ∈ S(Σ) be D-homogeneous polynomials, P such that deg(fj ) = [Lj ] for some line bundles L1 , . . . , Ls . Then F = sj=1 yj fj is homogeneous in R and its degree is the isomorphism class [OP(E) (1)] of the canonical line bundle on P(E) = P(L1 ⊕ · · · ⊕ Ls ). Proof. To prove that F is a homogeneous polynomial we will repeat the arguments in the proof of Lemma 3.5 in [CCD]. Let D1 , . . . , Dn be the torus-invariant divisors on P = PΣ corresponding to the 1-dimensional cones of the fan Σ. Then the pullback π ∗ Di is the torus-invariant divisor of P(E) corresponding to the cone generated by ẽi . Also denote by Dj0 the torusinvariant divisor corresponding to ñj . Let M̃ = M ⊕ M 0 be the lattice dual to Ñ = N ⊕ N 0 with M 0 = Hom(N 0 , Z) having {n∗2 , . . . , n∗s } as a basis dual ∗ to {n2 , . . . , ns }. The divisor corresponding to the character χnj is ∗ div(χnj ) = = n X i=1 n X hn∗j , ẽi iπ ∗ Di + s X hn∗j , ñk iDk0 k=1 (h1 (ei ) − hj (ei ))π ∗ Di − D10 + Dj0 . i=1 Therefore, [Dj0 ] + [π ∗ Lj ] all have the same degree in the Chow group Ad+s−2 (P(E)), and, consequently, F is a homogeneous polynomial. Now consider the following exact sequence [M]: 0 → OP(E) → π ∗ E ∗ ⊗ OP(E) (1) → TP(E) → π ∗ TP → 0, COHOMOLOGY OF COMPLETE INTERSECTIONS IN TORIC VARIETIES 137 where TX denotes the tangent bundle, E ∗ is the dual bundle. From here we can compute the Chern class c1 (TP(E) ) = c1 (π ∗ TP ) + c1 (π ∗ E ∗ ⊗ OP(E) (1)) = π ∗ c1 (TP ) − π ∗ c1 (E) + s · c1 (OP(E) (1)). Hence, s · c1 (OP(E) (1)) = π ∗ c1 (L1 ) + · · · + π ∗ c1 (Ls ) + c1 (TP(E) ) − π ∗ c1 (TP ). On the other hand, from the generalized Euler exact sequence [BC, §12] we get n−d 0 → OP → ⊕ni=1 OP (Di ) → TP → 0. This implies that c1 (TP ) = [D1 ] + · · · + [Dn ]. Similarly we have c1 (TP(E) ) = [π ∗ D1 ]+· · ·+[π ∗ Dn ]+[D10 ]+· · ·+[Ds0 ]. Under the identification Pic(P(E)) ,→ Ad+s−2 (P(E)) the first Chern class of a line bundle on P(E) is exactly its isomorphism class in the Picard group Pic(P(E)). Therefore s · [OP(E) (1)] = [π ∗ L1 ] + · · · + [π ∗ Ls ] + [D10 ] + · · · + [Ds0 ] = s · ([π ∗ L2 ] + [D20 ]). It can be easily checked that D20 is a Cartier divisor on P(E). Hence all classes [OP(E) (1)], [π ∗ L2 ] and [D20 ] lie in the Picard group Pic(P(E)). But this group is free abelian, because P(E) is complete. So the above equality is divisible by s: [OP(E) (1)] = [π ∗ L2 ] + [D20 ] = deg(F ). From now on we assume that P = PΣ is a complete simplicial toric variety and that deg(fj ) ∈ Pic(P), j = 1, . . . , s. Denote by Y the hypersurface in P P(E) defined by F = sj=1 yj fj . Lemma 2.2. X = Xf1 ∩ . . . ∩ Xfs is a quasi-smooth intersection iff the hypersurface Y is quasi-smooth. Proof. X = Xf1 ∩ . . . ∩ Xfs is a quasi-smooth intersection means that when ∂f ever x ∈ V(f1 , . . . , fs ) \ Z(Σ), the rank ∂xji (x) i,j = s. And Y is quasismooth iff z = (x, y) ∈ V(F ) \ Z(Σ̃) implies that one of the partial derivaP ∂f ∂F ∂F tives ∂y (z) = fj (x), j = 1, . . . , s, ∂x (z) = sj=1 yj ∂xji (x), i = 1, . . . , n, is j i nonzero. So let (x, y) ∈ V(F ) \ Z(Σ̃), then there is a cone σ̃ + σ 0 ∈ Σ̃ with σ ∈ Σ, Q Q σ 0 ∈ Σ0 , such that ẽi ∈σ̃ / xi ñj ∈σ / 0 yj 6= 0 where xi , yj are the coordinates of (x, y). If f (x) = · · · = f (x) = 0, then x ∈ V(f1 , . . . , fs ) \ Z(Σ) because 1 s Q ei ∈σ xi 6= 0. And if X = Xf1 ∩ . . . ∩ Xfs is a quasi-smooth intersection, one P ∂f ∂F (z) = sj=1 yj ∂xji (x), i = 1, . . . , n, is nonzero. of the partial derivatives ∂x i Conversely, suppose Y is quasi-smooth. Pick any x ∈ V(f1 , . . . , fs ) \ Z(Σ), then (x, y) ∈ V(F ) \ Z(Σ̃) for each y = (y1 , . . . , ys ) 6= 0. Therefore Ps ∂fj ∂fj j=1 yj ∂xi (x) 6= 0 for some i, which means the rank ∂xi (x) i,j is maximal. 138 ANVAR R. MAVLYUTOV 3. Cohomology of quasi-smooth intersections. Since a quasi-smooth intersection is a compact V-manifold (Proposition 1.3), the cohomology on it has a pure Hodge structure. Using Proposition 1.4 and the Poincaré duality, we can compute the cohomology of a quasi-smooth intersection except for the cohomology in the middle dimension d − s. So we introduce the following definition. d−s (X) is coker(H d−s (P) Definition 3.1. The variable cohomology group Hvar i∗ → H d−s (X)). The variable cohomology group also has a pure Hodge structure. Proposition 3.2. Let X = Xf1 ∩ . . . ∩ Xfs be a quasi-smooth intersection of ample hypersurfaces. Then there is an exact sequence of mixed Hodge structures ∪[X] d−s 0 → H d−s−1 (P) → H d+s−1 (P) → H d+s−1 (P \ X) → Hvar (X)→0, where [X] ∈ H 2s (P) is the cohomology class of X. Proof. Consider the Gysin exact sequence: (1) i i · · · → H i−2s (X) →! H i (P) → H i (P \ X) → H i−2s+1 (X) →! H i+1 (P) → · · · . d−s (X) is Since i∗ is Poincaré dual to the Gysin map i! , it follows that Hvar d−s d+s isomorphic to the kernel of i! : H (X) → H (P ). So we get an exact sequence i d−s H d−s−1 (X) →! H d+s−1 (P) → H d+s−1 (P \ X) → Hvar (X)→0. Now we use a commutative diagram i H d−s−1 (X) →! H d+s−1 (P) i∗ ↑ %∪[X] d−s−1 H (P). By Proposition 1.4 i∗ is an isomorphism in this diagram, so it suffices to prove that the Gysin map i! is injective in the above diagram. Lemma 3.3. If X = Xf1 ∩ . . . ∩ Xfs is a quasi-smooth intersection of ample i hypersurfaces, then the Gysin map H d−s−1 (X) →! H d+s−1 (P) is injective. Proof. Since the odd dimensional cohomology of a complete simplicial toric variety vanishes [F1, pp. 92-94] and i∗ : H d−s−1 (P) → H d−s−1 (X) is an isomorphism by Proposition 1.4, it follows that H d−s−1 (X) = H d−s−1 (P) = H d+s−1 (P) = 0 when d + s − 1 is odd. So by the Gysin exact sequence (1) it is enough to show that H d+s−2 (P \ X) = 0 when d + s − 2 is odd. To prove this we use the Cayley trick again. Let Y be the hypersurface defined by COHOMOLOGY OF COMPLETE INTERSECTIONS IN TORIC VARIETIES 139 P F = sj=1 yj fj . Then the natural map P(E) \ Y → P \ X, induced by the projection π : P(E) → P, is a Cs−1 bundle in the Zariski topology. Notice that P \ X is simply connected, because P is simply connected [F1, p. 56] and X has codimension at least 2 in P. Hence, the Leray-Serre spectral sequence implies that H i (P(E) \ Y ) = H i (P \ X) for i ≥ 0. We have that H d+s−2 (P(E)) = 0 for d + s − 2 odd and Y is quasi-smooth by Lemma 2.2. So from the Gysin exact sequence j H d+s−2 (P(E)) → H d+s−2 (P(E) \ Y ) → H d+s−3 (Y ) →! H d+s−1 (P(E)) (here the Gysin map j! is induced by the inclusion j : Y ,→ P(E)) it follows that we need to show injectivity of j! : H d+s−3 (Y ) → H d+s−1 (P(E)). Consider the commutative diagram j H d+s−3 (Y ) →! H d+s−1 (P(E)) j∗ ↑ %∪[Y ] d+s−3 H (P(E)) where [Y ] ∈ H 2 (P(E)) is the cohomology class of Y . The canonical line bundle OP(E) (1) is ample [H, III, §1], whence by Lemma 2.1, Y is ample. So by Proposition 10.8 [BC] j ∗ : H d+s−3 (P(E)) → H d+s−3 (Y ) is an isomorphism and by Hard Lefschetz ∪[Y ] : H d+s−3 (P(E)) → H d+s−1 (P(E)) is injective. Thus, from the above diagram the lemma follows. Definition 3.4. For a nonzero polynomial F ∈ R = C[x1 , . . . , xn , y1 , . . . , ys ] the Jacobian ring R(F ) denotes the quotient of R by the ideal generated by ∂F ∂F the partial derivatives ∂y , j = 1, . . . , s, ∂x , i = 1, . . . , n. j i Remark 3.5. If F = y1 f1 +· · ·+ys fs is as in Lemma 2.1 with fj ∈ Sαj , then R(F ) carries a natural grading by the Chow group Ad+s−2 (P(E)). Moreover, there are canonical isomorphisms Ad+s−2 (P(E)) ∼ = Ad−1 (P) ⊕ Ad (P) ∼ = Ad−1 (P) ⊕ Z ([F2]). With respect to this bigrading of the Chow group Ad+s−2 (P(E)) we have that deg(F ) = (0, 1), deg(fj ) = (αj , 0), deg(yj ) = (−αj , 1), which is very similar to the case when P is a projective space. We now can state the main result. Theorem 3.6. Let P be a d-dimensional complete simplicial toric variety, and let X ⊂ P be a quasi-smooth intersection of ample hypersurfaces defined by fj ∈ Sαj , j = 1, . . . , s. If F = y1 f1 + · · · + ys fs , then for p 6= d+s−1 2 , we have a canonical isomorphism R(F )(d+s−p)β−β ∼ = H p−s,d−p (X) 0 var where β0 = deg(x1 · · · xn · y1 · · · ys ), β = deg(F ) = deg(fj ) + deg(yj ). In the there is an exact sequence case p = d+s−1 2 d−s−1 d−s+1 , 2 ∪[X] 0 → H d−s−1 (P) → H d+s−1 (P) → R(F ) d+s+1 β−β0 → Hvar2 2 (X)→0. 140 ANVAR R. MAVLYUTOV Proof. Since H i (P) vanishes for i odd and has a pure Hodge structure of type (p, p) for i even, from Proposition 3.2 we get GrpF H d+s−1 (P \ X) ∼ = p−s,d−p d+s−1 Hvar (X) if p 6= d+s−1 , and in case p = the following sequence 2 2 ∪[X] 0 → H d−s−1 (P) → H d+s−1 (P) d+s−1 2 → GrF d−s−1 d−s+1 , 2 H d+s−1 (P \ X) → Hvar2 (X)→0 is exact. Now use the isomorphism of mixed Hodge structures H i (P \ X) ∼ = i H (P(E) \ Y ) and by the Theorem 10.6 [BC] the desired result follows. 4. Cohomology of nondegenerate intersections. In this section we consider a special case of quasi-smooth intersections. Definition 4.1. A closed subset X = Xf1 ∩. . .∩Xfs , defined by D-homogeneous polynomials f1 , . . . , fs , is called a nondegenerate intersection if Xfj1 ∩ . . . ∩ Xfjk ∩ Tτ is a smooth subvariety of codimension k in Tτ for any {j1 , . . . , jk } ⊂ {1, . . . , s} and τ ∈ Σ. (Here Tτ denotes the torus in PΣ associated with a cone τ ∈ Σ.) We will show how to define a nondegenerate intersection in terms of the polynomials f1 , . . . , fs . For σ ∈ Σ, let Uσ = {x ∈ An : x̂σ 6= 0}. We know that PΣ has an affine toric open cover by Aσ = Uσ /D(Σ), σ ∈ Σ [BC]. Also Tτ = (Uτ \ ∪γ≺τ Uγ )/D(Σ). Notice that Uτ \ ∪γ≺τ Uγ = {x ∈ An : x̂τ 6= 0, xi = 0 if ρi ⊂ τ } is a torus. So each Tτ is a quotient of a torus by a D-subgroup, because D is diagonalizable [BC]. Lemma 4.2. Let T = (C∗ )n /G be the quotient of a torus by a D-subgroup G. Suppose that X ⊂ (C∗ )n is an invariant subvariety with respect to the action of G. Then the geometric quotient X/G is smooth iff X is smooth. Proof. By the structure theorem of a D-group [Hu, §16.2] we can assume that (C∗ )n = G◦ × (C∗ )k , where G◦ ∼ = (C∗ )n−k is the identity component ◦ of G, and G = G × H for some finite subgroup H in (C∗ )k . Now it suffices to show the Lemma if G is a torus or a finite group. If G = G◦ then X = (C∗ )n−k × p(X), where by p(X) we mean the projection of X onto (C∗ )k . Notice that p(X) ∼ = X/G, hence X is smooth iff X/G is smooth. In the case G = H is a finite group it can be easily checked that X → X/G is an unramified cover [Sh, p. 346]. So X and X/G are smooth simultaneously. From this Lemma it follows that X = Xf1 ∩ . . . ∩ Xfs is a nondegenerate intersection iff V(fj1 , . . . , fjk ) ∩ Vτ is a smooth subvariety of codimension k in the torus Vτ = {x ∈ An : x̂τ 6= 0, xi = 0 if ρi ⊂ τ }. COHOMOLOGY OF COMPLETE INTERSECTIONS IN TORIC VARIETIES 141 AsP in Section 2 we can consider the hypersurface Y ⊂ P(E) defined by F = sj=1 yj fj . Lemma 4.3. X = Xf1 ∩ . . . ∩ Xfs is a nondegenerate intersection iff Y is a nondegenerate hypersurface. Proof. As shown above, X = Xf1 ∩ . . . ∩ Xfs is a nondegenerate intersection j∈{j ,...,jk } ∂f = k for all x ∈ V(fj1 , . . . , fjk ) ∩ Vτ , τ ∈ Σ if the rank ∂xji (x) i∈{i:1e ∈τ i/ } and {j1 , . . . , jk } ⊂ {1, . . . , s}. Similarly Y is nondegenerate iff z = (x, y) ∈ V(F ) ∩ Vτ̃ +τ 0 , τ̃ + τ 0 ∈ Σ̃ with τ ∈ Σ, τ 0 ∈ Σ0 (recall the definition of P(E) associated with Σ̃ in the Section 2) implies that one of the partial derivatives P ∂f ∂F ∂F / τ 0 }, ∂x (z) = sj=1 yj ∂xji (x), i ∈ {i : ẽi ∈ / τ̃ }, ∂yj (z) = fj (x), j ∈ {j : ñj ∈ i is nonzero. 0 0 0 Q Let (x,Qy) ∈ V(F ) ∩ Vτ̃ +τ 0 , where τ̃ + τ ∈ Σ̃ with τ ∈ Σ, 0τ ∈ Σ . Then ẽi ∈τ̃ / xi ñj ∈τ / 0 yj 6= 0 and xi = 0 if ẽi ∈ τ̃ , yj = 0 if ñj ∈ τ . If fj (x) = 0 for all j ∈ {j : ñj ∈ / τ 0 }, then x ∈ V(fj1 , . . . , fjk ) ∩ Vτ where {j1 , . . . , jk } = 0 {j : ñj ∈ / τ }. So if X = Xf1 ∩ . . . ∩ Xfs is a nondegenerate intersection, P ∂f ∂F (z) = sj=1 yj ∂xji (x), i ∈ {i : ẽi ∈ / τ̃ }, is one of the partial derivatives ∂x i nonzero. Conversely, suppose Y is nondegenerate. Take any x ∈ V(fj1 , . . . , fjk )∩Vτ with τ ∈ Σ, {j1 , . . . , jk } ⊂ {1, . . . , s}. Then (x, y) ∈ V(F ) ∩ Vτ̃ +τ 0 for each y ∈ Vτ 0 = {y ∈ As : yj 6= 0 if ñj ∈ / τ 0 , yj = 0 if ñj ∈ τ 0 } where 0 τ is the cone generated by the complement of {ñj1 , . . . , ñjk } in the set Ps ∂fj {ñ1 , . . . , ñs }. Therefore j=1 yj ∂xi (x) 6= 0 for some i, which means the j∈{j ,...,jk } ∂f rank ∂xji (x) i∈{i:1e ∈τ = k. / } i Since a nondegenerate hypersurface is quasi-smooth [BC], Lemma 2.2 shows that a nondegenerate intersection is quasi-smooth. Definition 4.4 ([BC]). Given a polynomial f ∈ S = C[x1 , . . . , xn ], we get the ideal quotient J1 (f ) = hx1 ∂f /∂x1 , . . . , xn ∂f /∂xn i : x1 · · · xn (see [CLO, p. 193]) and the ring R1 (f ) = S/J1 (f ). P Remark 4.5. If F = sj=1 yj fj ∈ R is as in Lemma 2.1, then R1 (F ) = R/J1 (F ) has a natural grading by the Chow group Ad+s−2 (P(E)) ∼ = Ad−1 (P) ⊕ Z. Theorem 4.6. Let X = Xf1 ∩ . . . ∩ Xfs be a nondegenerateP intersection of ample hypersurfaces given by fj ∈ Sαj , j = 1, . . . , s. If F = sj=1 yj fj ∈ R, then there is a canonical isomorphism p−s,d−p Hvar (X) = R1 (F )(d+s−p)β−β0 , where β0 = deg(x1 · · · xn · y1 · · · ys ), β = deg(F ). 142 ANVAR R. MAVLYUTOV Proof. First we will show that there is an isomorphism of Hodge structures d−s (X)(1 − s) ∼ H d+s−2 (Y ). Let ϕ : Y → P be the composition of the Hvar = var inclusion j : Y ,→ P(E) and the projection π : P(E) → P. As in [Te], consider the following morphism of the Leray spectral sequences E2p,q = 0 E p,q 2 H p (P, Rq π∗ C) ⇒ H p+q (P(E)) ↓ ↓ = H p (P, Rq ϕ∗ C) ⇒ H p+q (Y ). Since s−1 P if x ∈ X, Ps−2 if x ∈ / X, we have that (see [Go, p. 202], [De]) q if q is even and 0 ≤ q < 2s − 2, CP (− 2 ) Rq ϕ∗ C = CX (1 − s) if q = 2s − 2, 0 otherwise. ϕ−1 (X) = Also we have q R π∗ C = CP (− 2q ) 0 if q is even and 0 ≤ q ≤ 2s − 2, otherwise. The first spectral sequence degenerates at E2 , because for p or q odd Erp,q vanishes. The second spectral sequence also degenerates at E2 : hl−2s−2 (X) + 2s−4 X hl−q (P) = q=0 X p+q=l dim 0 E2p,q ≥ X p,q dim 0 E∞ = hl (Y ). p+q=l 0 E p,q 2 it suffices to show that the above inequality To show the degeneracy of is an equality. From Proposition 10.8 [BC] and Proposition 3.2 we get hd+s−2 (Y ) = hd+s−2 (P(E)) + hd+s−1 (P(E) \ Y ) − hd+s−1 (P(E)) + hd+s−3 (P(E)), hd−s (X) = hd−s (P) + hd+s−1 (P \ X) − hd+s−1 (P) + hd−s−1 (P). Hence, using the spectral sequence E2p,q , we can easily compute the Hodge P l−q (P) = hl (Y ) for numbers of P(E) and check that hl−2s−2 (X) + 2s−4 q=0 h l = d + s − 2. Using Proposition 1.4, we can similarly show the above equality for l 6= d + s − 2 as well. So the spectral sequence 0 E2p,q degenerates at E2 . Since E2d+s−2−q,q = 0 E2d+s−2−q,q for q 6= 2s − 2 and, by Proposition 1.4, E2d−s,2s−2 ,→ 0 E2d−s,2s−2 , we get an isomorphism of Hodge structures (for details see [Te]): d−s,2s−s d+s−2 d−s Hvar (Y ) ∼ /E2d−s,2s−2 ∼ (X)(1 − s). = 0 E2 = Hvar Now we only need to apply Theorem 11.8 [BC] to finish the proof. COHOMOLOGY OF COMPLETE INTERSECTIONS IN TORIC VARIETIES 143 References [BB] V.V. Batyrev and L. Borisov, Dual cones and mirror symmetry for generalized Calabi-Yau manifolds, in Mirror Symmetry II, Ed. B. Greene and S.-T. Yau, AMSIP, Providence, 1996. [BC] V.V. Batyrev and D.A. Cox, On the Hodge structure of projective hypersurfaces in toric varieties, Duke Math. J., 75 (1994), 293-338. [CCD] E. Cattani, D. Cox and A. Dickenstein, Residues in toric varieties, Compositio Mathematica, 108 (1997), 35-76. [C] D. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom., 4 (1995), 17-50. [CLO] D. Cox, J. Little and D. 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Libgober, Differential forms on complete intersections and related quotient module, Israel Mathematical Conference Proceedings, 9 (1996), 295-305. 144 ANVAR R. MAVLYUTOV [M] Yu. Manin, Lectures on the K-functor in algebraic geometry, Russ. Math. Surveys, 24(5) (1969), 1-89. [Na] J. Nagel, The Abel-Jacobi map for complete intersections, Indag. Math., 8 (1997), 95-113. [O] T. Oda, Convex Bodies and Algebraic Geometry, Springer-Verlag, Berlin, 1988. [PS] C. Peters and J. Steenbrink, Infinitesimal Variations of Hodge Structure and the Generic Torelli Problem for Projective Hypersurfaces (after Carlson, Donagi, Green, Griffiths, Harris), Classification of Algebraic and Analytic Manifolds, Ed. K. Ueno, Progr. Math., 39, Birkhäuser, Boston, (1983), 399-463. [Sh] I.R. Shafarevich, Basic Algebraic Geometry, Springer-Verlag, New York-Berlin, 1974. [St] J. Steenbrink, Intersection form for quasi-homogeneous singularities, Compositio Math., 34 (1977), 211-223. [Te] T. Terasoma, Infinitesimal variation of Hodge structures and the weak global Torelli theorem for complete intersections, Ann. of Math., 132 (1990), 213-235. Received February 18, 1998. University of Massachusetts Amherst, MA 01003 E-mail address: [email protected] PACIFIC JOURNAL OF MATHEMATICS Vol. 191, No. 1, 1999 CONNECTED SUMS OF SELF-DUAL MANIFOLDS AND EQUIVARIANT RELATIVE SMOOTHINGS Henrik Pedersen and Yat Sun Poon In this paper we address a problem in differential geometry using tools from algebraic geometry and the theory of singular complex spaces. We obtain examples of compact four dimensional self-dual conformal manifolds with torus symmetry and positive scalar curvature from twistor spaces with divisors and we study the local moduli of such geometries. 1. Introduction. Recall that a conformal 4-manifold is called self-dual if its Weyl curvature, considered as a bundle valued 2-form, is in the +1 eigenspace of the Hodge star-operator [1]. Due to Schoen’s proof [19] of the Yamabe conjecture it is known that within any conformal class on a compact manifold is a metric whose scalar curvature is constant and the sign of this constant is a conformal invariant. The main objective of this paper is to ensure that the scalar curvature is positive for the self-dual structures on the connected sums nCP2 of complex projective planes found in [13]. The metrics admit a torus T 2 of orientation preserving conformal isometries. For self-dual metrics the total space Z of the bundle of anti-self-dual 2forms of unit length is a complex 3-manifold. This complex manifold is the twistor space [1] and it gives an alternative description of self-duality. Indeed, Donaldson and Friedman [2] used a desingularisation of a singular model of the desired twistor space to prove existence of self-dual structures on nCP2 . The self-dual metric on CP2 is the Fubini-Study metric and the full moduli on 2CP2 had previously been obtained [15] via a different twistor construction. In [13] we adapted the theory of Donaldson and Friedman to obtain equivariant connected sums of compact self-dual manifolds. If the symmetry group is at least three-dimensional it is known [16] that the conformal metric is of non-negative type. In contrast Kim [7] obtained S 1 -symmetric examples of negative scalar curvature while LeBrun [11] gave examples on nCP2 of positive scalar curvature and with symmetry group S 1 . These metrics were obtained via an ansatz involving monopoles on hyperbolic 3-space. Similarly, Joyce [6] obtained T 2 -symmetric metrics on nCP2 of positive type using hyperbolic monopoles in two dimensions. These constructions give relatively easy access to knowledge about scalar curvature 145 146 H. PEDERSEN AND Y.S. POON while such insight is absent from our equivariant smoothings [13]. However, it is known [3] that if the complex manifold Z carries effective divisors, then the corresponding metric is of non-negative type and if futhermore the intersection form of the 4-manifold is positive definite the scalar curvature is in fact positive [10]. Kim and Pontecorvo [8] extended the work of Donaldson and Friedman obtaining a way of constructing scalar-flat Kähler surfaces based on relative complex deformations of singular twistor spaces with divisors of degree 1. We combine the equivariant and the relative smoothing programme to obtain a local moduli space of dimension n − 1 of T 2 -symmetric self-dual structures on nCP2 such that the associated twistor spaces all have smooth degree 1 and 2 divisors. In particular, all these self-dual metrics have positive scalar curvature. We also use our equivariant relative smoothing approach to obtain antiself-dual Hermitian metrics with non-semi-free circle action on the blow up 2 (S 1 × S 3 )#nCP , for n ≥ 3, of the Hopf suface in more than two points. This should be compared with LeBrun’s examples on the same spaces [12] but with semi-free circle symmetry. 2. The Geometrical Construction. Our study of self-duality on a 4-manifold M takes place on the associated twistor space Z. As a smooth manifold Z is the total space of the sphere bundle S(Λ2− ) → M of the bundle of anti-self-dual 2-forms. Equivalently, Z π is the total space of the bundle P (V − ) −→ M of projectivised half spinors. The twistor space is a complex 3-manifold and for any point x ∈ M the twistor line in Z, π −1 (x) = Lx , is a rational curve with normal bundle N the sum O(1) ⊕ O(1) of degree 1 line bundles. Invariantly N = H − ⊗ Vx+ where H − → P (Vx− ) is the hyperplane bundle. The antipodal map on the fibers of π : Z → M induces a real structure on Z [1]. The anti-canonical 1 bundle K −1 = Λ3,0 T Z of Z has a square root K − 2 which, when restricted onto a twistor line, coincides with the degree 2 holomorphic line bundle O(2). The zero set in the twistor space of a real holomorphic section of 1 K − 2 is called a degree 2 divisor. Such a divisor S may either be irreducible, in which case it is smooth [14], or it may decompose into a conjugate pair D, D̄ of smooth degree 1 divisors meeting in exactly one twistor line [17]. Now, assume we have two self-dual manifolds M1 and M2 with associated twistor spaces Z1 and Z2 . Furthermore, assume Z1 contains an irreducible degree 2 divisor S and let L1 = Lx1 be a twistor line above x1 ∈ M1 meeting S transversely in a conjugate pair of points q and q̄. Also, let D2 , D2 be a conjugate pair of divisors in Z2 meeting in the twistor line L2 = Lx2 above x2 ∈ M2 . Then, following Donaldson and Friedman [2], we blow up the twistor spaces Zi along the twistor lines Li , i = 1, 2. The proper transform CONNECTED SUMS OF SELF-DUAL MANIFOLDS 147 S̃ of S in the blow-up Z̃1 of Z1 meets the exceptional divisor Q1 in two rational curves C1 and C̄1 . The normal bundles NCS̃1 , NC̄S̃ relative to S̃ are 1 ˜ both equal to O(−1). In the blow-up Z̃ of Z the proper transforms D̃ , D̄ 2 2 2 2 of D2 , D̄2 , respectively, meet the exceptional divisor Q2 in a conjugate pair of ˜ rational curves C2 , C̄2 with normal bundles NCD̃22 = O(1) and NC̄D̄2 = O(1). 2 From an orientation reversing isometry φ : Tx1 M1 → Tx2 M2 we obtain a holomorphic isomorphism φ = φ+ × φ− : Q1 = P (Vx+1 ) × P (Vx−1 ) → Q2 = P (Vx−2 ) × P (Vx+2 ) and we construct the singular complex space Z = Z̃1 ∪φ Z̃2 with normal crossing singularity along Q = Q1 = φ−1 (Q2 ). In Z we have the singular divisor S with normal crossing singularities along C = C1 = φ−1 (C2 ) and C̄ = C̄1 = φ−1 (C̄2 ). Inspired by Kim and Pontecorvo [8] we now proceed to study smoothings of the pair (Z, S). The aim is to show that in some cases the connected sum has a twistor space with a degree 2 divisor. Let us introduce the necessary notation: If Y1 , . . . , Yn are smooth submanifolds of a compact complex manifold X we consider the sheaf ΘXY1 ,...,Yn of holomorphic vector fields on X which are tangent to Yi along Yi , i = 1, . . . , n. It is well known [9] that infinitesimal relative deformations are given by the first cohomology of this sheaf and that obstructions lie in the second cohomology. For a compact singular complex space X the deformation theory is described in terms of global extension groups TXi = Exti (ΩX , OX ) where ΩX is the sheaf of Kähler differentials and OX is the structure sheaf. These groups are computed from the sheaves τXi = Ext i (ΩX , OX ) using the local to global spectral sequence E2p,q = H p (τXq ) ⇒ TXp+q [2], [4]. Here TX0 is the Lie algebra of the group of automorphisms of X , the first order deformations lie in TX1 and obstructions are in TX2 . For normal crossings τX2 = 0 and the local to global spectral sequence is given as (2.1) 0 → H 1 (τX0 ) → TX1 → H 0 (τX1 ) → H 2 (τX0 ) → TX2 → H 1 (τX1 ). In the situations we are going to investigate, we have H 1 (τX1 ) = 0. Furthermore, we impose conditions implying the vanishing of H 2 (τX0 ) so that TX2 = 0 and the deformations are unobstructed and parametrised by TX1 . The subspace H 1 (τX0 ) corresponds to deformations for which the singularities remain locally a product. If the image of an element in the projection TX1 → H 0 (τX1 ) does not vanish the corresponding deformed space is smooth. Furthermore, for X equal to the singular twistor space Z the smoothing results in a twistor space with real structure [2]. The singular theory above will be applied to X = Z and X = S. To study deformations of the singular pair f : S ,→ Z we employ the theory of Ran i [18]: Let TZ|S denote the extension groups Exti (ΩZ|S , OS ) [8] with the local 148 H. PEDERSEN AND Y.S. POON p+q to global spectral sequence E2p,q = H p Ext q (ΩZ|S , OS ) where E2p,q ⇒ TZ|S . This gives the sequence 0 1 1 0 ). ) → H 2 (τZ|S → H 0 (τZ|S ) → TZ|S 0 → H 1 (τZ|S (2.2) i Furthermore, the groups TZ|S fit into a sequence 1 → Tf2 → TS2 ⊕ TZ2 → → Tf1 → TS1 ⊕ TZ1 → TZ|S (2.3) where Tfi is a natural derived functor and Tf1 gives infinitesimal deformations of the singular pair. Also, Tf0 is the Lie algebra of the symmetries of the pair (Z, S) and Tf2 is the obstruction space. For pairs (X , Y) of complex spaces we also have the sheaf τX0 Y of derivations of X preserving the ideal sheaf of Y. This sheaf of relative derivations coincide with ΘXY for pairs of smooth manifolds. From inclusions and restrictions we obtain the exact sequence 0 → τX0 Y → τY0 ⊕ τX0 → τX0 |Y → 0. (2.4) In the associated sequence on cohomology (2.5) H 1 (τX0 Y ) → H 1 (τY0 ) ⊕ H 1 (τX0 ) → H 1 (τX0 |Y ) → H 2 (τX0 Y ) the first term corresponds to deformations for which the singularities remain locally a product so H 1 (τX0 Y ) sits naturally inside Tf1 . Call the quotient space ∆. Then we have the short exact sequence 0 → H 1 (τX0 Y ) → Tf1 → ∆ → 0. (2.6) Now put together (2.1) with X = Z and X = S, (2.2), (2.3), (2.5) with X = Z and Y = S and (2.6) to get the following commutative diagram. (2.7) 0 x ? ? 0 ) H 2 (τZ|S x ? ? H 2 (τS0 ) ⊕ H 2 (τZ0 ) x ? ? β α ∆ x ? ? 1 ) −−−−→ −−−−→ H 0 (τS1 ) ⊕ H 0 (τZ1 ) −−−−→ H 0 (τZ|S x x ? ? ? ? Tf1 x ? ? −−−−→ TS1 ⊕ TZ1 x ? ? γ −−−−→ 1 TZ|S x ? ? −−−−→ 0 Tf2 −−−−→TS2 ⊕ TZ2 0 0 0 H 1 (τZS ) −−−−→ H 1 (τS0 ) ⊕ H 1 (τZ0 ) −−−−→ H 1 (τZ|S ) −−−−→H 2 (τZS ) x x x ? ? ? ? ? ? 0 0 0 The morphism α is the quotient of the morphism Tf1 → TS1 ⊕ TZ1 and the squares in the diagram are commutative by naturality. The row with the morphisms α and β needs more explanation: The sheaf τZ1 is supported on CONNECTED SUMS OF SELF-DUAL MANIFOLDS 149 Z̃2 Z̃1 = OQ [2]. Likewise, a ⊗ φ∗ NQ Q ⊆ Z and is the sheaf of sections of NQ 2 1 1 1 ∼ ∼ choice of trivialization τZ = OQ induces trivializations τS1 ∼ = OC C̄ and τZ|S = 0 1 0 0 0 1 0 ∼ OQ|S = OC C̄ . Then H (τS ) = H (OC ) ⊕ H (OC̄ ), H (τZ ) = H (OQ ), 1 ) = H 0 (O )⊕H 0 (O ) and β (a, b), c = (a−c, b−c) which certainly H 0 (τZ|S C C̄ is surjective. The kernel of β is hc, c, ci ∼ = C. Now we shall make the following assumptions which are satisfied in examples to be considered later: Assumption 2.1. In the case where the degree 2 divisor is irreducible we assume the vanishing of the following cohomology groups: H 2 (Zi , ΘZi ), i = 1, 2; H 2 (S, ΘS ); H 2 (D2 , ΘD2 ); H 2 (D̄2 , ΘD̄2 ); H 2 (Z1 , ΘZ1 S ) and H 2 (Z2 , ΘZ2 D2 D̄2 ). We shall show how these assumptions lead to the vanishing of H 2 (τS0 ), 0 ) and therefore from (2.1) we have the vanishing of the H 2 (τZ0 ) and H 2 (τZS obstruction spaces TZ2 and TS2 . We also see that ∆ is equal to the kernel of β. Now a diagram chasing gives the surjectivity of the morphism γ so Tf2 = 0. 0 ) of Therefore, any element ω in the complement of the hyperplane H 1 (τZS Tf1 gives smoothings of Z and S as ω maps to non-zero elements in H 0 (τS1 ) and H 0 (τZ1 ). Thus the twistor space of the connected sum M1 #M2 has a degree 2 divisor. Later we return to a calculation of the dimension of the space Tf1 of infinitesimal deformations. We shall also consider the situation where the degree 2 divisor S decomposes into a conjugate pair D1 , D̄1 of degree 1 divisors meeting in a twistor line L. We still blow up Z1 along a twistor line L1 intersecting D1 + D̄1 ˜ transversely in a conjugate pair of points. Also the proper transforms D̃1 , D̄ 1 meet Q1 in C1 and C̄1 . As before we construct the singular twistor space Z with the singular divisors D, D̄ having normal crossing singularities along C and C̄ respectively. The divisors D and D̄ meet in L. In this situation S is the singular space D ∪L D̄ with normal crossing singularities along L, C and C̄. We make the Assumption 2.2. In the case where the degree 2 divisor is reducible we assume the vanishing of the following cohomology groups: H 2 (Zi , ΘZi ), H 2 (Di , ΘDi ), H 2 (D̄i , ΘD̄i ), i = 1, 2; H 2 (Z1 , ΘZ1 S ) and H 2 (Z2 , ΘZ2 D2 D̄2 ). Furthermore, we get a diagram similar to (2.7) and as in the case where S is irreducible the smoothing is unobstructed. However, we do not control the singularities of S along L. Therefore, a smoothing of Z may produce an irreducible smooth degree 2 divisor. We shall discuss this problem at the end of Section 5. 150 H. PEDERSEN AND Y.S. POON 3. Symmetries. In this section we assume M1 and M2 have a torus group T 2 of orientation preserving conformal transformations with fixed points x1 and x2 respectively. Furthermore, the orientation reversing isometry φ : Tx1 M1 → Tx2 M2 , used to make the connected sum, is assumed to be an equivariant map. In this situation we proved in [13] that if the smoothing is unobstructed then also the equivariant smoothing is unobstructed. We refer to that paper for details. Now we want to impose the symmetries onto the relative smoothing described in Section 2. Recall [13] that we may choose metrics in M1 and M2 for which the torus acts as isometries. Therefore, the isotropy representation ι at the fixed points map into SO(4). Also, the symmetries lift to the twistor space to give real holomorphic automorphisms. If x1 is a fixed point in M1 , and L1 is the twistor line with normal bundle N , then the well known isomorphisms from twistor theory (Tx1 M1 )C = H 0 (L1 , N ) = Vx+1 ⊗ Vx−1 (3.1) are equivariant. This means the isotropy representation on the complexified tangent space coincide with the induced holomorphic action on the sections of the normal bundle and with the tensor product ι+ ⊗ ι− on Vx+1 ⊗ Vx−1 . Here ι± are defined modulo Z2 from the isotropy representation ι and the projections onto the factors of SU (2) × SU (2). Since the twistor line L1 intersects the divisor S transversely at two distinct points, we assume that S is reducible to D1 + D̄1 when we consider the twistor geometry in a neighbourhood of L1 . Now suppose the divisors Di , D̄i in Zi , i = 1, 2 are invariant. Then the points q = P (Vx−1 ) ∩ D1 and q̄ = P (Vx−1 )∩D̄1 are fixed by the action on Z1 . Therefore if v ∈ Vx−1 represents q we have Vx−1 = span(v)⊕span(v̄) as a decomposition into T 2 -invariant subspaces. Furthermore, from the definition of the complex structure on Z1 the ∗ horizontal space Tq D1 is equal to Vx+1 ⊗ span(v) with the action ι+ ⊗ ι∗− . We have C1 = P (Tq D1 ) and the normal bundle NCD̃11 is the universal subbundle of Tq D1 over C1 . On the other side we have L2 = P (Vx−2 ) and the normal bundles NCD̃22 = ˜ NLD22 , NC̄D̄2 = NLD̄22 . As L2 is equal to the transversal intersection D2 ∩ D̄2 2 we have NLZ22 = NLD22 ⊕ NLD̄22 and this is a decomposition into invariant subbundles. From twistor theory we have NLZ22 = Vx+2 ⊗ H − . Also, the torus action on Vx+2 decomposes, Vx+2 = E ⊕ Ē, and the only decomposition of Vx+2 ⊗ H − into invariant subspaces is (E ⊗ H − ) ⊕ (Ē ⊗ H − ). We may therefore assume that NLD22 = E ⊗ H − and NLD̄22 = Ē ⊗ H − . CONNECTED SUMS OF SELF-DUAL MANIFOLDS 151 The T 2 -equivariant isomorphism φ− : Vx−1 → Vx+2 induced by φ : Tx1M1 → Tx2 M2 can be assumed to satisfy φ− span(v) = E and φ− span(v̄) = Ē. Now we have D = D̃1 ∪φ+ D̃2 where φ+ : C1 = P (Tq D1 ) = P (Vx+1 ) → P (Vx−2 ) = C2 and similarly with D̄. Recall the fact [13] that H 0 (τZ1 )T 2 = H 0 (τZ1 ) ∼ = C Z̃2 Z̃1 1 ∗ which is true because τZ = NQ1 ⊗ φ (NQ2 ) = OQ and the action is trivial. 1 ) 0 1 1 ∼ 2 Also, H 0 (τZ|S T 2 = H (τZ|S ) = C because τZ is supported by Q and Q intersects S transversely along C and C̄. Likewise we shall need the following: Lemma 3.1. The sheaf τD1 is supported on the curve C and is trivial. All the sections are T 2 -invariant, i.e. H 0 (τD1 )T 2 = H 0 (τD1 ) = C. Similarly H 0 (τS1 )T 2 = H 0 (τS1 ) = C2 when the degree 2 divisor S is irreducible. Proof. Since the singularity is a normal crossing we have τD1 = NCD̃11 ⊗φ∗ NCD̃22 [2]. The curve C1 is the exceptional divisor of the blowing up of q in D1 so NCD̃11 = O(−1). Also, NCD̃22 is isomorphic to NLD22 so NCD̃22 = O(1). Thus τD1 = OC . Furthermore, the torus action is trivial on τD1 : the action on NCD̃11 is ι+ ⊗ ι∗− as this normal bundle is the universal subbundle of Tq D1 = ∗ Vx+1 ⊗ span(v) . On the other hand φ∗ NCD̃22 = φ∗ NLD22 = φ∗ (E ⊗ H − ) = ∗ (φ∗− E) ⊗ (φ∗+ H − ) = span(v) ⊗ H + = (span(v))∗ ⊗ U + = (NCD̃11 )∗ . The torus action on NLD22 = E ⊗ H − is ι+ ⊗ ι∗− . Therefore the action on φ∗ NLD22 is ι− ⊗ ι∗+ and we get the trivial action on τD1 = NCD̃11 ⊗ φ∗ NCD̃22 from the tensor product ι+ ⊗ ι∗− ⊗ ι− ⊗ ι∗+ . 4. Vanishing Theorems and the Unobstructed Case. We consider an irreducible degree 2 divisor S and address the problem of 0 ). This will lead to a proving the vanishing of H 2 (τZ0 ), H 2 (τS0 ) and H 2 (τZS 2 theorem on the unobstructed T -equivariant relative smoothing. First some technical lemmae: Lemma 4.1. Let Y ⊆ X be a smooth hypersurface in a complex manifold X. Then for any sufficiently small open set U ⊆ X we have H i (U, ΘX ) = 0 and H i (U, ΘXY ) = 0, i ≥ 1. Proof. We have H i (U, ΘX ) = 0, i ≥ 1, as ΘX is locally freeand H i (U, O) = 0 ¯ by the ∂-Poincaré Lemma. Similarly H i U ∩ Y, OY (NYX ) = 0, i ≥ 1. From the sequence Y 0 → ΘXY → ΘX → OY (NX )→0 152 H. PEDERSEN AND Y.S. POON we get H i (U, ΘXY ) = 0, i ≥ 2. We may assume U is a coordinate patch where the vanishing x1 = 0 of the first coordinate defines Y . Therefore 0 → H 0 (U, ΘXY ) → H 0 (U, ΘX ) → H 0 U ∩ Y, OY (NYX ) → 0 is exact and it follows that H 1 (U, ΘXY ) = 0. Lemma 4.2. Let bi : Z̃i → Zi be the blow-up of Zi along the real twistor line Li , i = 1, 2. Then for all j ≥ 0: H j (Z̃i , ΘZ̃i ) = H j (Zi , ΘZi Li ). Proof. We drop the subscript “i”. The direct image sheaf b∗ ΘZ̃ is isomorphic to ΘZL via the differential of b db : H 0 (b−1 (U ), ΘZ̃ ) → H 0 (U, ΘZL ) for U ⊆ Z open. Certainly db maps into H 0 (U, ΘZL ) if we can prove that any vector field on b−1 (U ) is tangential to Q along Q. But this follows 0 0 from the vanishing of H (Q, [Q]) = H CP1 × CP1 , O(1, −1) and from the sequence 0 → T Q → T Z̃|Q → [Q]|Q → 0. Conversely, let L be given in local coordinates (z1 , z2 , z3 ) by z2 = z3 = 0. Then, a vector field X = a1 ∂z∂ 1 + (a2 z2 + a3 z3 ) ∂z∂ 2 + (a4 z2 + a5 z3 ) ∂z∂ 3 in ∂ H 0 (U, ΘZL ) can be lifted to X̃ = a1 ∂z∂ 1 + (a2 z2 + a3 z2 `) ∂z∂ 2 + (a4 + a5 `) ∂` in H 0 (b−1 (U ), ΘZ̃ ) where ` = ``32 is a coordinate on b−1 (U ) = (z1 , z2 , z3 ), [`2 , `3 ] ∈ U × CP1 | `2 z3 = `3 z2 . The lemma now follows from the Leray spectral sequence [4] if we can prove that the sheaf Ri b∗ (ΘZ̃ ) vanishes for all i ≥ 1 where Ri b∗ (ΘZ̃ )(U ) = H i (b−1 (U ), ΘZ̃ ). Thus, consider the Leray cover V1 ∪ V2 of b−1 (U ) given by coordinates (z1 , z2 , `) and (z1 , z3 , 1` ) respectively. Then H i (b−1 (U ), ΘZ̃ ) = 0, i ≥ 2, and H 1 (b−1 (U ), ΘZ̃ ) = H 1 (V1 ∩ V2 , ΘZ̃ ) = H 1 (C3 \C2 , ΘZ̃ ) = 0 as ¯ H 1 (C3 \C2 , O) = 0 by the ∂-Poincaré lemma. 0 ). Let Z 0 , S 0 be the Now, let us study the cohomology groups H j (τZS normalizations of Z, S respectively and let also Q0 denote the disjoint union of Q1 and Q2 . Then we have the following sequence on Z (4.1) 0 0 → τZS → q∗ ΘZ 0 S 0 Q0 → i∗ ΘQC C̄ → 0 where q : Z 0 → Z is the identification map and i : Q ,→ Z is the inclusion. Lemma 4.3. For all j ≥ 0 we have: H j (Z, q∗ ΘZ 0 S 0 Q0 ) = H j (Z 0 , ΘZ 0 S 0 Q0 ) = H j (Z̃1 , ΘZ̃1 S̃Q1 ) ⊕ H j (Z̃2 , ΘZ̃ ˜ 2 D̃2 D̄ 2 Q2 ). CONNECTED SUMS OF SELF-DUAL MANIFOLDS 153 Proof. This follows from the Leray Spectral sequence if we can prove the vanishing of Ri q∗ ΘZ 0 S 0 Q0 , i ≥ 1, or equivalently we need the vanishing of H i (U1 , ΘZ̃1 S̃Q1 ) and H i (U2 , ΘZ̃ D̃ D̄ ˜ Q ), i ≥ 1 for small open sets Uk , k = 2 2 2 2 1, 2. On U1 , where U1 ∩ S̃ ∩ Q1 is non-empty, we have (4.2) π i Z̃1 ) −→ 0 0 → ΘZ̃1 S̃Q1 −→ ΘZ̃1 S̃ −→ OQ1 (NQ 1 which is exact by the transversality of Q1 and S̃: Choose coordinates (x1 , x2 , x3 ) such that x1 = 0 defines S̃ and x2 = 0 defines Q1 . Then on Z̃1 ) where r is the restriction onto Q1 U1 π = r ◦ dx2 : ΘZ̃1 S̃ → OQ1 (NQ 1 and the exactness is now easily seen. Then, from Lemma 4.1 and the sequence (4.2) we get H i (U1 , ΘZ̃1 S̃Q1 ) = 0, i ≥ 2 and as (4.2) is true on the level of presheafs we have also the vanishing for i = 1. The vanishing of H i (U2 , ΘZ̃ D̃ D̄ ), i ≥ 1, may be obtained as above by choosing U2 so ˜ 2 2 2 Q2 ˜ ). small that it meets only D̃ (or D̄ 2 2 Lemma 4.4. For all j ≥ 0, H j (Z̃1 , ΘZ̃1 S̃Q1 ) = H j (Z1 , ΘZ1 SL1 ). Proof. As in the proof of Lemma 4.2 we have b1∗ ΘZ̃1 S̃Q1 = ΘZ1 SL1 . The result follows from the Leray spectral sequence once we have established the vanishing of Ri b1∗ (ΘZ̃1 S̃Q1 ) for i ≥ 1. We consider the sequences (4.2) and (4.3) 0 → ΘZ̃1 S̃ → ΘZ̃1 → OS̃ (NS̃Z̃1 ) → 0. Assume q̄ ∈ / U ⊆ Z1 and that the bundles are trivial on U . Then on b−1 1 (U ) Z̃1 −1 −1 ∗ ∗ we have OS̃ (NS̃ ) = [S̃]|S̃ = b1 [S]|S̃ ⊗ Q1 |S̃ = b1 [S]|S̃ ⊗ C1 . Thus on b−1 1 (U ) ∩ S̃ 0 → OS̃ (NS̃Z̃1 ) → b∗1 [S] → OC1 (b∗1 [S]) → 0 ∗ j 0 is exact. Also, H j (b−1 1 (U ) ∩ C1 , b1 [S]) = H (q, [S]) = 0 for j ≥ 1, H (q, [S]) = −1 j ∗ j [S]q = C and H (b1 (U ) ∩ S̃, b1 [S]) = H (U ∩ S, [S]) = 0, for j ≥ 1, by the ¯ ∂-Poincaré Lemma. Thus, H j b−1 (U ) ∩ S̃, O (N Z̃1 ) = 0, j ≥ 1, if we can 1 prove surjectivity of S̃ S̃ ∗ 0 −1 ∗ H 0 (b−1 1 (U ) ∩ S̃, b1 [S]) → H (b1 (U ) ∩ C1 , b1 [S]). This corresponds via b1 to the map H 0 (U ∩S, [S]) → H 0 (q, [S]) which is onto by evaluation. From the proof of Lemma 4.2 we also have H j (b−1 1 (U ), ΘZ̃1 ) = −1 j 0, j ≥ 1. Thus, from (4.3), H (b1 (U ), ΘZ̃1 S̃ ) = 0, j ≥ 1, once we have proved that Z̃1 0 −1 H 0 (b−1 1 (U ), ΘZ̃1 ) → H (b1 (U ) ∩ S̃, OS̃ (NS̃ )) 154 H. PEDERSEN AND Y.S. POON is onto. However, it is not hard to describe the vector fields on b−1 1 (U ) in local coordinates on patches V1 , V2 as in the proof of Lemma 4.2 and to see that the map is surjective. Z̃1 Now, as H j b−1 1 (U ), OQ1 (NQ1 ) = 0, j ≥ 1, we get from (4.2) that H j (b−1 1 (U ), ΘZ̃1 S̃Q1 ) = 0, j ≥ 2. The map Z̃1 0 −1 H 0 (b−1 1 (U ), ΘZ̃1 S̃ ) → H b1 (U ), OQ1 (NQ1 ) is onto which again is seen using local coordinates on b−1 1 (U ). Thus we have Ri b∗ (ΘZ̃1 S̃Q1 ) = 0, i ≥ 1 and the lemma is proved. Similarly we have: Lemma 4.5. For all j ≥ 0, H j (Z̃2 , ΘZ̃ ˜ 2 D̃2 D̄ 2 Q2 Proof. Again b2∗ ΘZ̃ ˜ 2 D̃2 D̄ 2 Q2 ) = H j (Z2 , ΘZ2 D2 D̄2 ). = ΘZ2 D2 D̄2 via the differential db2 . Furthermore Ri b ) vanishes for i ≥ 1: First we prove the vanish˜ 2∗ (ΘZ̃ D̃ D̄ 2 2 2 Q2 −1 j ing of H (b2 (U2 ), ΘZ̃ D̃ D̄ ˜ ), j ≥ 1, U2 ⊆ Z2 . Use coordinates (z1 , z2 , `), 2 2 2 1 (z1 , z3 , ` ) on a cover V1 ∪ V2 of b−1 2 (U2 ) such that D is given by z2 = 0, the sheaf D̄ by z3 = 0 and L2 by z2 = z3 = 0. Lemma 4.1 gives that this is a Leray cover of b−1 ˜ (V2 ) = ΘZ̃2 D̃2 (V2 ) and D̃2 in V2 is 2 (U2 ) because ΘZ̃2 D̃2 D̄ 2 3 2 a smooth hypersurface. Also, ΘZ̃ D̃ D̄ ˜ (V1 ∩ V2 ) = ΘZ̃2 (C \C ). Therefore H j (b−1 2 (U2 ), ΘZ̃ 2 ˜ 2 D̃2 D̄ 2 2 2 ) vanishes, j ≥ 1. Then consider the sequence Z̃2 → OQ2 (NQ ) → 0. 2 Z̃2 The vanishing of H j b−1 2 (U2 )∩Q2 , OQ2 (NQ2 ) , j ≥ 1 is seen using the Leray cover V1 , V2 . Also, using coordinates on V1 , V2 we can show that vector fields may be lifted so we have surjectivity of Z̃2 H 0 b−1 → H 0 b−1 ˜ 2 (U2 ), ΘZ̃ D̃ D̄ 2 (U2 ), OQ2 (NQ2 ) . 0 → ΘZ̃ ˜ 2 D̃2 D̄ 2 Q2 2 → ΘZ̃ ˜ 2 D̃2 D̄ 2 2 2 Then the short exact sequence gives H j (b−1 ˜ Q ) = 0, j ≥ 1, 2 (U2 ), ΘZ̃2 D̃2 D̄ 2 2 and we have proved the lemma. To make use of the sequence (4.1) we prove the following lemma. Lemma 4.6. The cohomology groups H j (Q, ΘQC C̄ ) vanishes for j ≥ 1 and H 0 (Q, ΘQC C̄ ) = su(V + ) ⊕ u(1). Proof. H j (Q, ΘQ) = 0, j ≥ 1 and H 0 (Q, ΘQ ) = H 0 P (Vx+1 ), O(2) ⊕ H 0 P (Vx−1 ), O(2) = S 2 (Vx+1 )∗ ⊕ S 2 (Vx−1 )∗ = su(Vx−1 ) ⊕ su(Vx−1 ). From the sequence (4.4) 0 → ΘQC C̄ → ΘQ → OC C̄ (NCQC̄ ) → 0 CONNECTED SUMS OF SELF-DUAL MANIFOLDS 155 and the vanishing of H j C, OC (NCQ ) , j ≥ 1 we get H 2 (Q, ΘQC C̄ ) = 0. Now, H 0 (Q, ΘQ ) is generated by global SU (2) × SU (2) holomorphic transformations. Elements in H 0 (Q, ΘQC C̄ ) are the vector fields generated by actions leaving C and C̄ invariant, i.e. by rotations on L1 leaving q and q̄ fixed. Thus H 0 (Q, QC C̄ ) is generated by SU (Vx+1 ) × U (1) where U (1) ≤ SU (Vx−1 ). Also, the map H 0 (Q, ΘQ ) → H 0 C, OC (NCQ ) ⊕ H 0 C̄, OC̄ (NC̄Q ) is surjective so H 1 (Q, ΘQC C̄ ) = 0. We are now able to prove the vanishing of the relative obstruction group. Proposition 4.7. Assume H 2 (Z1 , ΘZ1 S ) = 0 and H 2 (Z2 , ΘZ2 D2 D̄2 ) = 0. 0 ) vanishes. Then the obstruction group H 2 (τZS Proof. The long exact sequence associated to (4.1) together with Lemmae 4.3, 4.4, 4.5 and 4.6 gives 0 H 2 (Z, τZS ) = H 2 (Z1 , ΘZ1 SL1 ) ⊕ H 2 (Z2 , ΘZ2 D2 D̄2 ). The second summand vanishes due to the assumption. Consider the exact sequence 0 → ΘZ1 SL1 → ΘZ1 S → OL1 (NLZ11 ) → 0. We have H 1 L1 , OL1 (NLZ11 ) = 0 and also H 2 (Z1 , ΘZ1 S ) = 0 so the first summand also vanishes. (4.5) As explained in Section 2 we also need the following vanishing results. Proposition 4.8. Assume H 2 (Zi , ΘZi ) = 0, i = 1, 2, H 2 (S, ΘS ) = 0 and H 2 (D2 , ΘD2 ) = 0. Then the obstruction groups H 2 (τZ0 ) and H 2 (τS0 ) vanishes. Proof. Consider the normalization in the following sequence 0 → τZ0 → q∗ ΘZ 0 Q0 → i∗ ΘQ → 0. (4.6) Then, from Lemma 4.1 and the Leray spectral sequence, we get H 2 (Z, q∗ ΘZ 0 Q0 ) = H 2 (Z̃1 , ΘZ̃1 Q1 ) ⊕ H 2 (Z̃2 , ΘZ̃2 Q2 ). From the sequence Z̃i 0 → ΘZ̃i Qi → ΘZ̃i → OQi (NQ )→0 i Z̃i and the vanishing of H j Qi , OQi (NQ ) , j ≥ 1, we have H 2 (Z̃i , ΘZ̃i Qi ) = i H 2 (Z̃i , ΘZ̃i ) which is equal to H 2 (Zi , ΘZi Li ) by Lemma 4.2. Since Zi j 2 2 H Li , OLi (NLi ) = 0, j ≥ 1, we have H (Zi , ΘZi Li ) = H (Zi , ΘZi ), i = 1, 2, and since H j (Qi , ΘQi ) = 0, i = 1, 2, j ≥ 1, the long exact sequence 156 H. PEDERSEN AND Y.S. POON associated to (4.6) gives H 2 (τZ0 ) = H 2 (Zi , ΘZ1 ) ⊕ H 2 (Z2 , ΘZ2 ) which vanishes by assumptions. This was just a repetition of the result of Donaldson and Friedman [2]. Next we concentrate on the divisor. Again we have a normalization sequence (4.7) 0 → τS0 → q∗ ΘS 0 C 0 C̄ 0 → i∗ ΘC C̄ → 0. Lemma 4.1 together with the Leray spectral sequence give ˜ ,Θ H 2 (S, q∗ ΘS 0 C 0 C̄ 0 ) = H 2 (S̃, ΘS̃C1 C̄1 ) ⊕ H 2 (D̃2 , ΘD̃2 C2 ) ⊕ H 2 (D̄ ˜ C̄ ). 2 D̄ 2 2 1 Then, as H j C1 , OC1 (NCS̃1 ) = H j CP , O(−1) = 0, j ≥ 0, the sequence 0 → ΘS̃C1 C̄1 → ΘS̃ → OC1 (NCS̃1 ) ⊕ OC̄1 (NC̄S̃1 ) → 0 gives H j (S̃, ΘS̃C1 C̄1 ) = H j (S̃, ΘS̃ ), j ≥ 0. Essentially by repeating the arguments in Lemma 4.2 we get H j (S̃, ΘS̃ ) = H j (S, ΘSqq̄ ), j ≥ 0. For dimensional reasons we have H j q q̄, Oqq̄ (NqSq̄ ) = 0, j ≥ 1, so H 2 (S, ΘSqq̄ ) = H 2 (S, ΘS ). For D̃2 we have N D̃ ∼ = O(1) so the sequence C2 0 → ΘD̃2 C2 → ΘD̃2 → OC2 (NCD̃22 ) → 0 gives H 2 (D̃2 , ΘD̃C2 ) = H 2 (D̃2 , ΘD̃2 ). Then as ΘC2 = O(2) the long exact sequence associated to (4.7) gives H 2 (τS0 ) = H 2 (S, ΘS ) ⊕ H 2 (D2 , ΘD2 ) ⊕ H 2 (D̄2 , ΘD̄2 ) which vanishes by assumptions. This ends our proof of the fact that under the general Assumption 2.1, the twistor space of the connected sum M1 #M2 has an irreducible degree 2 divisor. Finally we want to summarize and at the same time bring the symmetries back into considerations. Theorem 4.9. Let M1 , M2 be compact self-dual conformal 4-manifolds with torus symmetry and fixed points. Assume the isotropy representations at the fixed points are intertwined via an orientation reversing isometry. If the Assumption 2.1 is satisfied for T 2 -invariant divisors S, D2 and D̄2 , then there is a complex equivariant smoothing of the singular twistor space Z and the singular divisor S into a twistor space with torus action and an invariant irreducible degree 2 divisor. Proof. Due to the equivariance of the various sheaf morphisms we get an analogue of diagram (2.7) where the morphisms are betweeen the G-invariant part of the groups. Consider the vanishing results in Propositions 4.7 and 4.8. The spectral sequence (2.1) now gives the vanishing of the obstruction spaces TZ2 and TS2 . Then, chasing the diagram (2.7) gives surjectivity of the CONNECTED SUMS OF SELF-DUAL MANIFOLDS 157 morphism γ so Tf2 = 0. Therefore, any element ω in the complement of 0 ) of T 1 gives smoothings of Z and S as ω maps to the hyperplane H 1 (τZS f non-zero elements in H 0 (τS1 ) and H 0 (τZ1 ). Thus we obtain a smooth twistor space with an irreducible degree 2 divisor. Together with Lemma 3.1 this shows that there are T 2 -equivariant deformations of Z and S. 0 ) = 0 if we assume that each compoRemark. We can also prove H 2 (τZ|S nent Mi , i = 1, 2, has positive scalar curvature. As this vanishing result is not used in this paper, we only give a short outline of the proof: Start with the normalization sequence (4.8) 0 → q∗ ΘZ 0 Q0 |S → i∗ ΘQ|C C̄ → 0. 0 → τZ|S Then, as (q∗ ΘZ 0 Q0 )|S = q∗ (ΘZ 0 Q0 |S 0 ), we get H j S, q∗ ΘZ 0 Q0 |S ˜ ,Θ = H j S̃, ΘZ̃1 Q1 |S̃ ⊕ H j D̃2 , ΘZ̃2 Q2 |D̃2 ⊕ H j D̄ 2 Z̃ ˜ 2 Q2 |D̄ 2 , j ≥ 0. We study each component in this sum. From the sequence Z̃1 0 → ΘZ̃1 Q1 |S̃ → ΘZ̃1 |S̃ → OC1 C̄1 NQ (4.9) → 0, 1 Z̃1 and the vanishing H j S̃, OC1 C̄1 (NQ ) = 0, j ≥ 0, we get H j (S̃, ΘZ̃1 Q1 |S̃ ) = 1 H j (S̃, ΘZ̃1 |S̃ ), j ≥ 0. Consider therefore the sequence (4.10) 0 → ΘS̃ → ΘZ̃1 |S̃ → OS̃ NS̃Z̃1 → 0. As NS̃Z̃1 = b∗1 [S]|S̃ ⊗ [C1 ]−1 ⊗ [C̄1 ]−1 and OC1 C̄1 (b∗1 [S]) = OC1 C̄1 we have (4.11) 0 → NS̃Z̃1 → b∗1 [S]|S̃ → OC1 C̄1 → 0. 1 Also, H j (S̃, b∗1 [S]) = H j (S, [S]) = H j (S, K − 2 ), j ≥ 0. Then, due to the positive scalar curvature and the vanishing results of Hitchin [5], the sequence 1 1 0 → OZ1 → K − 2 → OS (K − 2 ) → 0 gives H 2 (S̃, b∗1 [S]) = 0. Then (4.11) gives H 2 S̃, OS̃ (NS̃Z̃1 ) = 0. From the sequence (4.10) and from H 2 (S̃, ΘS̃ ) = H 2 (S, ΘSqq̄ ) = H 2 (S, ΘS ) = 0 we get H 2 (S̃, ΘZ̃1 |S̃ ) = 0. Thus H 2 (S̃, ΘZ̃1 Q1 |S̃ ) = 0. To prove the vanishing of H 2 (D̃2 , ΘZ̃2 Q2 |D̃2 ) we proceed as above with sequences similar to (4.9), (4.10), (4.11) and the use of vanishing results due to the positive scalar curvature of M2 . Thus we have H 2 (S, q∗ ΘZ 0 Q0 |S ) = 0 and (4.8) then gives 0 ) = 0. H 2 (S, τZ|S 158 H. PEDERSEN AND Y.S. POON 5. A Local Moduli Space of Self-Dual Metrics on nCP2 . We shall continue our study [13] of the local moduli of T 2 -symmetric selfdual structures on the connected sums of n copies of the complex projective plane. As in [13] it is assumed that the only orbits are tori on which T 2 acts freely, circles stabilized by some S 1 -subgroup inside T 2 and isolated fixed points. We construct nCP2 from CP2 by attaching planes step by step and at the same time keeping divisors and symmetries. However, let us take the general approach a little further in the case of an irreducible degree 2 divisor. From the equivariant version of (2.7) we get the sequence 0 0 → H 1 (τZS )T 2 → (Tf1 )T 2 → ∆T 2 → 0. (5.1) Similarly (4.1) and Lemmae 4.3, 4.4, 4.5 and 4.6 give the sequence (5.2) 0 0 → H 0 (τZS )T 2 → H 0 (Z1 , ΘZ1 SL1 )T 2 ⊕ H 0 (Z2 , ΘZ2 D2 D̄2 )T 2 → su(V + ) ⊕ u(1) T 2 0 → H 1 (τZS )T 2 → H 1 (Z1 , ΘZ1 SL1 )T 2 ⊕ H 1 (Z2 , ΘZ2 D2 D̄2 )T 2 → 0. Furthermore, from (4.5) we have the sequence (5.3) 0 → H 0 (Z1 , ΘZ1 SL1 )T 2 → H 0 (Z1 , ΘZ1 S )T 2 → H 0 L1 , O(NLZ11 ) T 2 → H 1 (Z1 , ΘZ1 SL1 )T 2 → H 1 (Z1 , ΘZ1 S )T 2 → 0. Thus, putting (5.1), (5.2) and (5.3) together, we get a formula for the dimensions: Let χ1 (FX )T 2 = h0 (X, FX )T 2 − h1 (X, FX )T 2 for a sheaf FX on a complex manifold X. Proposition 5.1. Let (Tx1 M1 )T 2 denote the real vector space of T 2 -invariant tangent vectors at x1 and let C(T 2 ) denote the Lie algebra of the centralizer of T 2 in SO(3) × SO(3). Then dim(Tf1 )T 2 0 = 1 + h1 (τZS )T 2 0 = 1 + h0 (τZS )T 2 + dimR (Tx1 M1 )T 2 + dimR C(T 2 ) −χ1 (ΘZ1 S )T 2 − χ1 (ΘZ2 D2 D̄2 )T 2 . Proof. We have used the Kodaira equivalence (3.1) between sections of the normal bundle of the line L1 in the twistor space and tangent vectors at the corresponding point x1 in M1 . The symmetry group T 2 sits in SO(3)×SO(3) via the composition of the isotropy representation i : T 2 → SO(4) and the representation of SO(4) on Λ2+ ⊕ Λ2− . Now we need a series of lemmae which will ensure that we have the general Assumption 2.1 satisfied and will make it possible to find the number dim(Tf1 )T 2 in the case of irreducible degree 2 divisors in the connected sums of complex projective planes. CONNECTED SUMS OF SELF-DUAL MANIFOLDS 159 Lemma 5.2. Let Z2 be the twistor space of CP2 . Then H 2 (Z2 , ΘZ2 ) = 0, H 2 (D2 , ΘD2 ) = 0, H 2 (D̄2 , ΘD̄2 ) = 0, H 0 (Z2 , ΘZ2 D2 D̄2 ) ∼ = u(1) ⊕ su(2) C ⊆ su(3)C and H 1 (Z2 , ΘZ2 D2 D̄2 ) = H 2 (Z2 , ΘZ2 D2 D̄2 ) = 0. Proof. The space H 0 (Z2 , ΘZ2 ) is generated by the complexifications SU (3)C of the lifts of the isometries on CP2 to the flag n o X (5.4) Z2 = ([v0 , v1 , v2 ], [`0 , `1 , `2 ]) | vi `i = 0 . Then H 0 (Z2 , ΘZ2 D2 D̄2 ) consists of the vector fields generated by the group that leaves D2 ∪ D̄2 invariant. For D2 given by v0 = 0 and D̄2 by `0 = 0 it is clear that H 0 (Z2 , ΘZ2 D2 D̄2 ) must contain and therefore be equal to the maximal subalgebra u(1) ⊕ su(2) C of su(3)C . Since Z2 is a flag manifold and D2 is the blow-up of CP2 at one point, H 2 (Z2 , ΘZ2 ) and H 2 (D2 , ΘD2 ) vanish. Z̃2 Lemma 4.5 and the vanishing of H j Q2 , OQ2 (NQ ) , j ≥ 0, give H j (Z2 , 2 ΘZ2 D2 D̄2 ) = H j (Z̃2 , ΘZ̃ D̃ D̄ ) = H j (Z̃2 , ΘZ̃ D̃ D̄ ˜ ˜ ). Consider the sequence 2 2 2 Q2 2 2 2 0 → ΘZ̃ D̃ D̄ (5.5) N Z̃2 ˜ → 0. ˜ → ΘZ̃2 → OD̃ D̄ ˜ 2 Here Hj 2 2 Z̃2 Z̃2 , OD̃ D̄ ˜ (N ˜ 2 2 D̃2 D̄ 2 2 2 D̃2 D̄2 ) is equal to the sum of H j (D2 , [D2 ] ⊗ [L2 ]−1 ) Z̃2 and the conjugate part. This follows because ND̃ = b∗2 [D2 ]|D̃2 ⊗ [C2 ]−1 2 and b2 gives isomorphisms D̃2 ∼ = D2 , C2 ∼ = L2 . Identify D2 to the blow2 up of CP at one point with hyperplane class H and exceptional divisor E, then D2 |D2 = H − E and D̄2 |D2 = L2 = H [17]. It follows that H j (D2 , [D2 ]⊗[L2 ]−1 ) vanishes for j ≥ 0 as H 0 (D2 , −E) = 0, H 0 (D2 , −3H + 2E) = 0 and χ(−E) = 0 by the Riemann-Roch formula. The sequence j (5.5) now shows that H j (Z̃2 , ΘZ̃ D̃ D̄ ˜ ) = H (Z̃2 , ΘZ̃2 ) which coincide with 2 2 2 H j (Z2 , ΘZ2 L2 ) as shown in Lemma 4.2. Tracing these identities we now have H j (Z2 , ΘZ2 D2 D̄2 ) = H j (Z2 , ΘZ2 L2 ), j ≥ 0. As H j L2 , OL2 (NLZ22 ) = 0, j ≥ 1, the sequence (5.6) 0 → ΘZ2 L2 → ΘZ2 → NLZ22 → 0 gives H 2 (Z2 , ΘZ2 L2 ) = H 2 (Z2 , ΘZ2 ) which vanishes for the flag. Thus, H 2 (Z2 , ΘZ2 D2 D̄2 ) = 0 as claimed. Also, as D2 ∩ D̄2 = L2 , H 0 (Z2 , ΘZ2 L2 ) contains H 0 (Z2 , ΘZ2 D2 D̄2 ) = u(1) ⊕ su(2) C , which is maximal, so H 0 (Z2 , ΘZ2 L2 ) is 4-dimensional. Since H 0 L2 , OL2 (NLZ22 ) is 4-dimensional so (5.6) gives H 1 (Z2 , ΘZ2 L2 ) = H 1 (Z2 , ΘZ2 ) which vanishes for the flag. Thus we get H 1 (Z2 , ΘZ2 D2 D̄2 ) = 0 and the lemma has been proved. Consider the twistor space (5.4) of CP2 with the action of T 2 given by [v0 eiΦ1 , v1 eiΦ2 , v2 e−i(Φ1 +Φ2 ) ], [`0 e−iΦ1 , `1 e−iΦ2 , `2 ei(Φ1 +Φ2 ) ] 160 H. PEDERSEN AND Y.S. POON where Φi = pi θ+qi ψ, i = 1, 2, (θ, ψ) ∈ T 2 and the integers pi , qi satisfy P p1 q 2 − p2 q1 = ±1 [13]. Then the irreducible degree 2 divisor S given by λi vi `i = 0, where λi 6= λj , i 6= j, is T 2 -invariant. Indeed T 2 is exactly the symmetry group of S. The twistor fibration onto CP2 is [z0 , z1 , z2 ] = [v1 `¯2 −v2 `¯1 , v2 `¯0 − v0 `¯2 , v0 `¯1 − v1 `¯0 ] and it induces the action [z0 e−iΦ1 , z1 e−iΦ2 , z2 ei(Φ1 +Φ2 ) ] on CP2 . Take x = [0, 0, 1] ∈ CP2 . Then the twistor line Lx is given by v2 = `2 = 0 with two fixed points q = [0, 1, 0], [1, 0, 0] and q̄ = [1, 0, 0], [0, 1, 0] . A generic T 2 -invariant degree 2 divisor is transversal to Lx and contains q and q̄. We get a reducible invariant divisor with D1 given by v0 = 0 and D̄1 given by `0 = 0. We have Lx ∩ D1 = q and Lx ∩ D̄1 = q̄. Finally take D2 given by v2 = 0 and D̄2 corresponding to `2 = 0. Then Lx = D2 ∩ D̄2 . Thus we have the following building blocks. Lemma 5.3. The twistor space Z of CP2 contains T 2 -invariant divisors S, Di , D̄i , i = 1, 2 as in Section 2 with arbitrary isotropy representation at fixed points and satisfying H 0 (Z, ΘZS )T 2 = C2 , H 2 (S, ΘS ) = 0, H 2 (Di , ΘDi ) = 0, H 2 (D̄i , ΘD̄i ) = 0 and H 2 (Z, ΘZDi D̄i ) = 0. Proof. The vanishing of H 2 (S, ΘS ) follows because H 2 (Z, ΘZ ) = 0 and because S is contained in Z with positive normal bundle. H 0 (Z, ΘZS )T 2 contains at least the algebra generated by T 2 as S is invariant and for the generic S described above where λi 6= λj , i 6= j, the symmetry group can at most be two dimensional. With Lemma 5.2 and the discussion above in mind, the lemma is proved. Note that in order to fulfill all the vanishing conditions stated in Assumptions 2.1 and 2.2 we still need to prove H 2 (Z, ΘZS ) = 0 for the flag manifold Z. This could be done, mutatis mutandis, as for ΘZDD̄ in Lemma 5.2. However, as we focus on the T 2 -equivariant situation we only prove the vanishing of the T 2 -invariant part of this cohomology group. Indeed it follows from the next more general lemma. Lemma 5.4. Let Z1 be the twistor space of nCP2 and let S ⊆ Z1 be an irreducible degree 2 divisor with canonical bundle KS . Then H 0 (S, O(KS−1 ))T 2 ∼ = C, H 1 (S, O(KS−1 ))T 2 = 0 and H 2 (Z1 , ΘZ1 S )T 2 = 0. Proof. It is known [14] that S is the blow-up Sn of a real quadric S0 n times in a pair of conjugate points. Let bk : Sk → Sk−1 , 1 ≤ k ≤ n, be the blow-down map from the blow-up of S0 k times to the blow-up k − 1 times. The torus action has only isolated fixed points on Sn so the action on Sk , 1 ≤ k ≤ n, induced by blowing down, also only has isolated fixed points. Therefore, the points qk−1 , q̄k−1 of blowing-up from Sk−1 to Sk is at the intersection of invariant divisors. For topological reasons the action on S0 has four isolated fixed points. CONNECTED SUMS OF SELF-DUAL MANIFOLDS 161 Let Kk = KSk and let Ek , Ēk be the exceptional divisor of the blowing-up −1 bk . Then Kk−1 = b∗k K(k−1) ⊗ Ek−1 ⊗ Ēk−1 and −1 −1 )→0 0 → O(Kk−1 ) → O(b∗k K(k−1) ) → OEk ∪Ēk (b∗k K(k−1) is exact and induces −1 0 → H 0 (Sk , O(Kk−1 ))T 2 → H 0 Sk , O(b∗k K(k−1) ) 2 T 0 ∗ −1 → H Ek ∪ Ēk , O(bk K(k−1) ) 2 T −1 −1 1 1 → H (Sk , O(Kk ))T 2 → H Sk , O(b∗k K(k−1) ) 2 → 0. T −1 −1 −1 0 ∗ Now, H Ek ∪ Ēk , O(bk K(k−1) ) is isomorphic to K(k−1),qk−1 ⊕ K(k−1),q̄ . k−1 −1 2 0 ∗ Furthermore, the T -action on H Ek ∪ Ēk , O(bk K(k−1) ) is given as nontrivial rotations on each factor: Let qk−1 ∈ Sk−1 and let A, B be two invariant divisors such that qk−1 ∈ A ∩ B and such that in local coordinates (z1 , z2 ), A is given by z2 = 0 and B is given as z1 = 0. As the T 2 -action does not have 2-dimensional fixed point set but only isolated fixed points, we may assume the action on Tqk−1 Sqk−1 = Tqk−1 A ⊕ Tqk−1 B is given as −1 = Λ2 Tqk−1 Sqk−1 the action is ei(Φ1 +Φ2 ) . This diag (eiΦ1 , eiΦ2 ). On Kk−1,q k−1 action will not be trivial due to the assumption p1 q2 − p2 q1 = ±1. On the blow-up the action is (eiΦ1 , ei(Φ2 −Φ1 ) ) near the intersection of the exceptional divisor and the proper transform à and near B̃ it is (eiΦ2 , ei(Φ1 −Φ2 ) ). Therefore the weights must be linear independent at all the points qk−1 . −1 0 ∗ Thus H Ek ∪ Ēk , O(bk K(k−1) ) T 2 vanishes. By an inductive argument we get from (5.7) that H j S, O(KS−1 ) T 2 = H j S0 , O(K0−1 ) T 2 , j = 0, 1. As S0 is a quadric surface we have H 1 S0 , O(K0−1 ) = 0. Choose coordinates ([s0 , s1 ], [t0 , t1 ]) on S0 = CP1 × CP1 such that the action on S0 is [s0 , s1 eiΦ1 ], [t0 , t1 eiΦ2 ] . The nine monomials (s20 t20 , . . . , s21 t21 ) of order four give a basis of H 0 S0 , O(K0−1 ) = H 0 CP1 × CP1 , O(2, 2) . The weights of the induced T 2 -action vanishes on s20 t20 and is of the form nΦ1 + mΦ2 on the eight other monomials. We claim that s20 t20 is the only invariant section: Note that if the weights (Φ1 , Φ2 ) of the isotropy representation at [1, 0], [1, 0] ∈ S0 are linearly dependent then the weights of the isotropy representations at the other three fixed points on S0 are also linearly dependent. Since the weights at the fixed points of the T 2 -action on the blow-up are linear combinations of Φ1 and Φ2 the weights on S will be linearly dependent too. It follows, by restricting the twistor projection onto S, that on nCP2 there is a fixedpoint where p1 q2 − p2 q2 = 0 which is a contradiction. Thus, H 0 S, O(KS−1 ) T 2 ∼ = C. Finally, as [S] = KS−1 , the sequence (5.7) (5.8) 0 → ΘZ1 S → ΘZ1 → OS (KS−1 ) → 0, 162 H. PEDERSEN AND Y.S. POON gives H 2 (Z1 , ΘZ1 S )T 2 = 0. We are now able to find the number dim(Tf1 )T 2 in a concrete example. Proposition 5.5. It is possible to construct a T 2 -symmetric twistor space of nCP2 with invariant irreducible degree 2 divisors by equivariant relative smoothings using CP2 as building blocks and proceeding step by step. Furthermore, in the final step we have dim(Tf1 )T 2 = h1 (Z, ΘZS )T 2 = n, where Z is the smooth twistor space of nCP2 . Proof. The assumptions which give unobstructed equivariant relative smoothings of two planes are satisfied. This follows from Lemmae 5.2, 5.3 and 5.4. Using upper semi-continuity of the dimension of cohomology we may indeed proceed step by step to get T 2 -symmetric twistor spaces of nCP2 with degree 2 irreducible divisors. With the assumption, p1 q2 − p2 q1 = ±1, on the T 2 -action we have dimR (Tx1 M1 )T2 = 0 and dimR C(T 2) = 2. Also, from Lemma 5.2, we get χ1 (ΘZ2 D2 D̄2 )T 2 = dim u(1) ⊕ su(2) T 2 = 2. Lemma 5.3 and upper semi0 ) continuity gives h0 (Z1 , ΘZ1 S )T 2 = h0 (τZS T 2 = 2 in each step. From the sequence (5.8) and Lemma 5.4 we get (5.9) 0 → H 0 S, O(KS−1 ) T 2 → H 1 (Z1 , ΘZ1 S )T 2 → H 1 (Z1 , ΘZ1 )T 2 → 0 and h0 S, O(KS−1 ) T 2 = 1. In [13] we got h1 (Z1 , ΘZ1 )T 2 = n − 2 if Z1 corresponds to (n − 1)CP2 , so h1 (Z1 , ΘZ1 S )T 2 = n − 1. Now, putting all these data into the formula in Proposition 5.1, we get dim(Tf1 )T 2 = n which by upper semi-continuity coincide with h1 (Z, ΘZS )T 2 for the smooth twistor space Z of nCP with smooth irreducible degree 2 divisor S. Thus, we may summarize and formulate the following result concerning the local moduli space of self-dual structures on nCP2 constructed in [13]. Theorem 5.6. Consider the local moduli space of T 2 -symmetric self-dual structures on nCP2 obtained by equivariant smoothings. The self-dual structures are all associated to T 2 -symmetric twistor spaces Z with invariant irreducible degree 2 divisors. In particular, the scalar curvature of each Yamabe metric is positive. For n ≥ 3, the T 2 -action can be chosen such that all S 1 subgroups are non-semi-free. The dimension of the local moduli is equal to h1 (Z, ΘZ )T 2 = n − 1. Proof. Note that we have ensured that all twistor spaces have symmetry group of dimension 2 so a local moduli space is well-defined. To prove this theorem, we need to prove that the deformation is target stable [18]. By taking the second components of the maps from the first CONNECTED SUMS OF SELF-DUAL MANIFOLDS 163 column to the second column in diagram (2.7), we have 0 ) 1 0 −−−→ H 1 (τZS T 2 −−−→ (Tf )T 2 −−−→ εy δy ∆T 2 α̂y −−−→ 0 0 −−−→ H 1 (τZ0 )T 2 −−−→ (TZ1 )T 2 −−−→ H 0 (τZ1 )T 2 −−−→ 0 The target stability is equivalent to δ being surjective. Since α̂ is the identity map, δ is surjective if ε is surjective. By Lemma 5.3 and upper semi-continuity, the exact sequence (5.2) gives the isomorphism ∼ = 0 H 1 (τZS )T 2 −→ H 1 (Z1 , ΘZ1 SL1 )T 2 . Also (5.3) gives the isomorphism ∼ = H 1 (Z1 , ΘZ1 SL1 )T 2 −→ H 1 (Z1 , ΘZ1 S )T 2 . Similarly, one has the isomorphisms [13] ∼ = ∼ = H 1 (τZ0 )T 2 −→ H 1 (Z1 , ΘZ1 L1 )T 2 and H 1 (Z1 , ΘZ1 L1 )T 2 −→ H 1 (Z1 , ΘZ1 )T 2 . Since the two compositions ∼ = 0 H 1 (τZS )T 2 −→ H 1 (Z1 , ΘZ1 S )T 2 and ∼ = H 1 (τZ0 )T 2 −→ H 1 (Z1 , ΘZ1 )T 2 are induced by normalizations and restrictions, the inclusions induce a commutative diagram 0 ) H 1 (τZS T2 k H 1 (Z1 , ΘZ1 S )T 2 ε −→ H 1 (τZ0 )T 2 k λ −→ H 1 (Z1 , ΘZ1 )T 2 . By (5.9), λ is surjective. Therefore ε is surjective. Thus, the local moduli space obtained in [13] does indeed correspond to twistor spaces carrying divisors as claimed. Therefore the corresponding Yamabe metrics are of non-negative type [3]. In fact the scalar curvature must be positive because the intersection form for nCP2 is positive definite [10]. The fact that we may assume the S 1 subgroups are all non-semi-free was proved in [13]. Turning to the problem of finding degree 1 divisors we first prove: Lemma 5.7. If L is the twistor line over a fixed point of a T 2 -symmetric nCP2 and if S is an invariant degree 2 divisor containing L, then S is reducible. 164 H. PEDERSEN AND Y.S. POON Proof. Assume on the contrary that S is irreducible. Then S is the blowup of a quadric surface with L a smooth fiber [14]. In particular it does not pass through any points of blowing-up and this implies that L is not invariant: On the quadric the only T 2 -invariant curves are the two conjugate pair of generator lines passing through the four fixed points which are the only points of blowing-up as S has T 2 -symmetry. Proposition 5.8. Given a T 2 -symmetric twistor space over nCP2 with an invariant irreducible divisor S. Let L be a twistor line above an isolated fixed point. Then there exists a conjugate pair D, D̄ of invariant degree 1 divisors intersecting along L. Proof. Let V1 , V2 be a pair of independent holomorphic vector fields generated by the torus action on the twistor space Z. Since S is invariant the restrictions of V1 and V2 to S are tangential to S. Therefore we have an invariant section ŝ = V1 ∧ V2 |S ∈ H 0 (S, KS−1 ). We claim that this section is nontrivial. From Lemma 5.7 L intersects S transversely at q, q̄. Since the T 2 -action only has isolated fixed points, there exist complex coordinates (z1 , z2 ) centered at q such that the T 2 -action is given as (einθ z1 , eimψ z2 ) with n and m both non-zero. Therefore ŝ = nmz1 z2 ∂z∂ 1 ∧ ∂z∂ 2 near q. Since H 0 (S, KS−1 ) is non-trivial and h1 (Z, O) = h1 (nCP2 , R) = 0 [5], the exact sequence 1 0 → O → K − 2 → OS (KS−1 ) → 0 1 implies that the section ŝ lifts to an invariant section of K − 2 . 1 Let s ∈ H 0 (Z, K − 2 ) be the section such that s−1 (0) = S. Then the 1 subspace V = span{s, ŝ} in H 0 (Z, K − 2 ) is T 2 -invariant. Every element of this system contains q and q̄ and we may choose an element S 0 ∈ |V| containing p ∈ L where p 6= q and p 6= q̄. As S 0 is of degree 2 it also contains L. We may assume S 0 is real. The divisor S 0 is invariant, otherwise as L is invariant, L would be contained in the base locus of the system |V| but S does not contain L. Then by Lemma 5.7, S 0 must be reducible. Corollary 5.9. The twistor spaces described in Theorem 5.6 contain T 2 invariant reducible degree 2 divisors. Remark. Given a T 2 -symmetric twistor space over nCP2 with an invariant reducible degree 2 divisor as above, we construct T 2 -symmetric singular twistor spaces with degree 2 divisors over (n + 1)CP2 as outlined at the end of Section 2. As in Section 4, mutatis mutandis, we can prove that the obstructions to the equivariant relative smoothing vanish provided Assumption 2.2 is satisfied. The cohomology group H 2 (Zi , ΘZi ) vanishes by upper semicontinuity. The groups H 2 (Di , ΘDi ), H 2 (D̄i , ΘD̄i ), i = 1, 2; H 2 (Z1 , ΘZ1 S )T 2 and H 2 (Z2 , ΘZ2 D2 D̄2 )T 2 is proved to vanish as outlined at the end of the CONNECTED SUMS OF SELF-DUAL MANIFOLDS 165 paper. However, as we do not control the singularity of the degree 2 divisors in the smoothing process, we cannot conclude a priori that there are reducible degree 2 divisors after the smoothing. Furthermore, the notion of local moduli becomes dubious when the degree 2 divisor is reducible. We do not pursue this issue. However, the existence of reducible degre 2 divisors is secured by Corollary 5.9. This has applications as shown in the next section. 6. Anti-Self-Dual Hermitian Surfaces. In [13] we constructed self-dual metrics on (S 1 ×S 3 )#nCP2 , n ≥ 3, such that the symmetry group is S 1 and the action is non-semi-free. We begin with a CP2 with S 1 -action [z0 eiθ , z1 , z2 ]. We label the fixed points as P0 = [0, 1, 0], P1 = [1, 0, 0] and A1 = [0, 0, 1]. Then we attach a CP2 with S 1 -action [z0 e−iθ , z1 , z2 ] and fixed points A2 = [0, 0, 1], P2 = [1, 0, 0], P∞ = [0, 1, 0]. We attach A1 to A2 via an orientation reversing isometry and of course using the smoothing of the twistor space. Note that the isotropy representation of P0 is (eiθ x, y) while near P∞ it is (e−iθ x, y). Therefore, we may consider the possibility of making a self sum by identifying P0 and P∞ . Note that at this stage the S 1 -action is semi-free but if we equivariantly attach a CP2 to P2 we get a S 1 -symmetric 3CP2 with non-semi-free action and we can still make a self-sum at P0 and P∞ . Indeed we can go on attaching more CP2 -blocks away from P0 and P∞ and then make a self-sum at P0 and P∞ to obtain self-dual structures on (S 1 × S 3 )#nCP2 , n ≥ 3, with non-semi-free S 1 -symmetry. To get a complex structure on (S 1 × S 3 )#nCP2 compatible with the conformal metric but with opposite orientation we bring in the relative smoothing: Due to Corollary 5.9, we may assume that on nCP2 we have a non-semi-free S 1 -symmetric twistor space with a reducible divisor D + D̄ such that D ∩ D̄ = LP0 and such that the twistor line above P∞ intersect D and D̄ transversely. Then make an equivariant self-sum at P0 , P∞ relative to the divisor D + D̄. Following the notation from Section 2 we have curves ˜ above P . Also, there are curves C , C̄ ˜ above P which are C1 , C̄1 in D̃, D̄ ∞ 2 2 0 both mapped onto LP0 by the blowing-down. Theorem 6.1. There exist anti-self-dual conformal Hermitian metrics on 2 (S 1 × S 3 )#nCP , for n ≥ 3, such that the symmetry group is S 1 and the action is non-semi-free. Proof. We claim the isotropy data ensures that C1 in D̃ is identified to C2 in ˜ : On 2CP2 we have the four fixed points P , P , P , P . D̃ and not to C̄2 in D̄ 0 1 2 ∞ 166 H. PEDERSEN AND Y.S. POON We draw the spheres between points as lines and get a diagram: P∞ −1 0 1 P2 1 (6.1) 0 P0 −1 1 −1 P1 Here the numbers represent the weights of the S 1 -action along the spheres near the fixed points. These numbers are easily obtained from the information above about the S 1 action. The twistor space of CP2 was described in (5.4). We see that the twistor line LP0 is given by v1 = 0 = `1 and the divisor corresponds to v1 = 0 and its conjugate corresponds to `1 = 0. On LP0 the S 1 -action has the two fixed points q0 = ([0, 0, 1], [1, 0, 0]) and q̄0 = ([1, 0, 0], [0, 0, 1]). On the divisor the isotropy data is given by q̄0 1 0 (6.2) −1 q0 −1 Now, the divisor D in the twistor space of 2CP2 is the blow up of CP2 twice [17]. It has the following configuration q̄0 q∞ 0 1 L P0 −1 q0 q2 −1 (6.3) q1 CONNECTED SUMS OF SELF-DUAL MANIFOLDS 167 with π(qi ) = pi where π is the restriction of the twistor projection π : D → 2CP2 . Note that π maps the whole line LP0 to P0 and otherwise is an orientation reversing equivariant diffeomorphism. Therefore, by comparing with the data (6.1) on 2CP2 , we obtain the full isotropy data on D: q̄0 0 0 1 q∞ 1 −1 −1 q0 1 −1 1 (6.4) q2 −1 q1 From the real structure σ : D → D̄ we can obtain the isotropy data on D̄ near q0 , q̄0 , q̄1 , q̄2 , q̄∞ . Then, as in the proof of Lemma 5.4 we get the data 168 H. PEDERSEN AND Y.S. POON on the blow-up of q∞ : on D̃ we get a 0 0 α 1 1 C2 C1 −1 −1 b β 1 −1 1 −1 −1 (6.5) 1 ˜ we get and on D̄ ā 0 0 ᾱ −1 −1 C̄2 C̄1 1 1 b̄ β̄ 1 −1 1 −1 (6.6) 1 −1 Here the points a, b̄ are mapped to q¯0 by the blow-down and ā, b are mapped to q0 . The points α, ᾱ corresponds to q∞ , q̄∞ while β, β̄ correspond to q2 , q¯2 . For the sake of completeness we may compute the isotropy data ˜ : At at a, b, α, β, ā, b̄, ᾱ, β̄ also in directions transversal to the surfaces D̃, D̄ CONNECTED SUMS OF SELF-DUAL MANIFOLDS 169 each point let (z, v, w) denote coordinates with (v, w) coordinates along the exceptional divisor, the quadric surface, and with (z, v) coordinates along the blown up divisor. Then with respect to such coordinates the weights of the isotropy is a(0, 1, 1), b(−1, −1, 1), α(0, 1, 1), β(1, −1, 1) and ā(0, −1, −1), b̄(1, 1, −1), ᾱ(0, −1, −1), β̄(−1, 1, −1). The identification map (z, v, w) 7→ (z̄, v, w) gives the orientation change on the divisor and it is now clear that a is identified to α, and b to β. This shows that we need to attach C2 to C1 and C̄2 to C̄1 . For nCP2 , in a neighbourhood of the invariant sphere joining the fixed points P0 and P∞ , the isotropy data is identical to the 2CP2 case. Therefore, the arguments above can be applied to prove the claim also for nCP2 . ˜ are glued to themselves respectively. Therefore, It follows that D̃ and D̄ the singular divisor S in the singular twistor space Z is a disjoint union of two degree 1 divisors D and D̄. In the next paragraph, we prove that the obstructions to smoothing the pair (Z, S) vanish. The resulting smooth twistor space carries a conjugate pair of disjoint degree 1 divisors intersecting all twistor lines transversely. This pair corresponds to complex structures ±I on (S 1 ×S 3 )#nCP2 compatible with the metric but inducing the opposite orientation. To prove that the obstructions to the S 1 -equivariant smoothing of the pair (Z, S) vanish, we proceed as in Section 4. It suffices to prove the vanishing of H 2 (Z, ΘZDD̄ )S 1 and H 2 (D, ΘD )S 1 . By Serre duality, h2 (D, ΘD ) = h0 (D, KD ⊗ Ω1 ). Since D is T 2 -invariant, −1 , where V1 and V2 is a pair of (V1 ∧ V2 )|D is a non-trivial section of KD independent vector fields generated by the torus action on the twistor space. Therefore, if h0 (D, KD ⊗ Ω1 ) was not equal to zero, there would have been non-trivial holomorphic 1-forms on D. Since D is a rational surface, it follows that H 2 (D, ΘD ) vanishes. To deal with H 2 (Z, ΘZDD̄ )S 1 , we consider the following exact sequences Z 0 → ΘZDD̄ → ΘZ D̄ → OD (ND ) → 0, and Z 0 → ΘZ D̄ → ΘZ → OD̄ (ND̄ ) → 0. By upper semi-continuity, H 2 (Z, ΘZ ) vanishes. With the next lemma, we conclude that H 2 (Z, ΘZDD̄ )S 1 vanishes. Lemma 6.2. Let Z be the twistor space of nCP2 with T 2 symmetry and let Z )) D ⊆ Z be an invariant degree 1 divisor. Then H j (D, O(ND S 1 = 0 and Z j H (D̄, OD̄ (ND̄ )) = 0, j ≥ 1. Proof. Recall [17] that D is the blow-up of CP2 n times. For 1 ≤ k ≤ n consider the sequence bk : Dk → Dk−1 of blowing down to D0 = CP2 . Let pk−1 ∈ Dk−1 be the point which is blown up and let Ek be the exceptional 170 H. PEDERSEN AND Y.S. POON P divisor. On D = Dn , [Dn ] denotes the line bundle of H − ni=1 Ei and on P Dk , [Dk ] is the line bundle of H − ki=1 Ei [17]. Then [Dk ] = b∗k [Dk−1 ]⊗Ek−1 and from 0 → [Dk ] → b∗k [Dk−1 ] → OEk (b∗k [Dk−1 ]) → 0 we get (6.7) 0 → H 0 (Dk , [Dk ])S 1 → H 0 (Dk−1 , [Dk−1 ])S 1 → ([Dk−1 ]pk−1 )S 1 → H 1 (Dk , [Dk ])S 1 → H 1 (Dk−1 , [Dk−1 ])S 1 → 0 and H 2 (Dk , [Dk ]) = H 2 (Dk−1 , [Dk−1 ]). Thus by induction we have Z1 H 2 (D, O(ND )) = H 2 (Dn , [Dn ]) = H 2 (D0 , [D0 ]) = H 2 (CP2 , [H]) = 0. Now we prove ([Dk ]pk )S 1 = 0, 0 ≤ k ≤ n − 1. Let gk be the blow-down from Dk to CP2 . Then on Dk we have [17] −1 KD = [D̄k ]|Dk ⊗ [Dk ]|Dk = (gk∗ H)2 ⊗ [Dk ]|Dk k −1 so (KD )pk = (gk∗ H)2pk ⊗ [Dk ]pk . In general, we may assume that the action k on CP2 is [z0 , z1 eiΦ1 , z2 eiΦ2 ] in homogeneous coordinates. Then for p = gk (pk ), (gk∗ H)3pk = Hp3 ∼ = Λ2 Tp CP2 so the weight of (gk∗ H)2pk is 23 (Φ1 + Φ2 ). Since pk is on the intersection of invariant curves Tpk Dk has weight (n1 Φ1 + −1 )pk has weight mΦ1 +nΦ2 for some integers n2 Φ2 , m1 Φ1 +m2 Φ2 ). Then (KD m, n. From diagram (6.4), after the curves joining q1 to q2 , and q2 to q∞ are blown down, we see that Φ1 = 0 and Φ2 = 1. Therefore, the representation on [Dk ]pk is non-trivial. This gives ([Dk ]pk )S 1 = 0, 0 ≤ k ≤ n − 1, and then by induction (6.7) gives H 1 (D, [D])S 1 = 0. Theorem 6.1 should be compared with LeBrun’s examples [12] of S 1 symmetric anti-self-dual Hermitian metrics on blow-up of Hopf surfaces. Note that the S 1 -action in his examples is semi-free. References [1] M.F. Atiyah, N.J. Hitchin and I.M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London, A362 (1978), 425-461. [2] S.K. Donaldson and R. Friedman, Connected sums of self-dual manifolds and deformations of singular spaces, Nonlinearity, 2 (1989), 197-239. [3] P. Gauduchon, Structures de Weyl et théorèmes d’annulation sur une variété conforme autoduale, Norm. Sup. Pisa, 18 (1991), 563-629. [4] R. Hartshorne, Algebraic Geometry, GTM 52, 1977, Springer-Verlag, New York. [5] N.J. Hitchin, Linear field equations on self-dual spaces, Proc. Roy. Soc. London, Sér A, 370 (1980), 173-191. [6] D. Joyce, Explicit construction of self-dual 4-manifolds, Duke Math. J., 77 (1995), 519-552. CONNECTED SUMS OF SELF-DUAL MANIFOLDS 171 [7] J. Kim, On the scalar curvature of self-dual manifolds, Math. Ann., 297 (1993), 235-251. [8] J. Kim and M. Pontecorvo, A new method of constructing scalar-flat Kähler surfaces, J. Differential Geometry, 41 (1995), 449-477. [9] K. Kodaira and D. Spencer, On deformations of complex analytic structure, I, II, III, Ann. of Math., 67 (1958), 328-466; 71 (1960), 43-76. [10] C. LeBrun, On the topology of self-dual 4-manifolds, Proc. Amer. Math. Soc., 98 (1986), 637–640. [11] , Explicit self-dual metrics on CP2 # · · · #CP2 , J. Differential Geometry, 34 (1991), 223-253. [12] , Anti-self-dual Hermitian metrics on blown-up Hopf surfaces, Math. Ann., 289 (1991), 383-392. [13] H. Pedersen and Y.S. Poon, Equivariant connected sums of compact self-dual manifolds, Math. Ann., 301 (1995), 717-749. [14] , Self-duality and differentiable structures on the connected sum of complex projective planes, Proc. Amer. Math. Soc., 121 (1994), 859-864. [15] Y.S. Poon, Compact self-dual manifolds with positive scalar curvature, J. Differential Geometry, 24 (1986), 97-132. [16] , Conformal transformations of compact self-dual manifolds, International J. Math., 5 (1994), 125-140. [17] , On the algebraic structure of twistor spaces, J. Differential Geometry, 36 (1992), 451-491. [18] Z. Ran, Stability of certain holomorphic maps, J. Differential Geometry, 34 (1991), 37–47. [19] R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geometry, 20 (1984), 478-495. Received January 23, 1998. This project is partially supported by the NSF grant DMS9504908. Odense University DK-5230 Odense M Denmark E-mail address: [email protected] University of California, Riverside Riverside, CA 92521 E-mail address: [email protected] PACIFIC JOURNAL OF MATHEMATICS Vol. 191, No. 1, 1999 SOME SUMMATIONS OF q-SERIES BY TELESCOPING M.V. Subbarao and A. Verma Summation formulae for q-series with independent bases are obtained and used to derive transformation and expansion of q-series involving independent bases. 1. Introduction. The sum of the first (n + 1)-terms of the non-terminating very-well-poised 6 φ5 [q][1] (1.1) n a X ; q)k (aq, bq, cq, aq (1 − aq 2k ) (a, b, c, bc bc ; q)n q k = aq aq aq aq (1 − a) (q, b , c , bcq; q)k ( b , c , bcq, q; q)n k=0 follows from Jacksons q-analgue of Whipple’s summation formula for a terminating very-well-poised balanced 8 φ7 [q] (in [4, 2.6.2] setting e = aq n+1 ). A bibasic analogue of (1.1) was obtained by Gasper [3]. n a X ; q)k (1 − aq k pk )(1 − bpk q −k ) (a, b; p)k (c, bc qk aq ap (1 − a)(1 − b) (q, b , q)k ( c , bcp; p)k (1.2) k=0 = (ap, bp; p)n (cq, aq bc ; q)n , aq ap (q, b ; q)n ( c , bcp; p)n n = 0, 1, 2, . . . and he used it for obtaining quadratic and cubic summation and transformations formulae for q-hypergeometric series. A little later Gasper and Rahman [5] obtained a bilateral extension of Gasper’s bibasic summation formula (1.2): If m, n are non-negative integers, then (1.3) 2 n bpk q −k X (c, ad (1 − adpk q k ) (1 − d ) (a, b; p)k bc ; q)k qk b adq adp bcp (1 − ad) (1 − ) (dq, ; q) ( , ; p) k k d b c d k=−m = × (1 − a)(1 − b)(1 − c)(1 − d(1 − ad)(1 − db )(1 − dc )(1 ( ad2 q (ap, bp; p)n (cq, bc ; q)n ad2 bc ) − ad bc ) adp bcp (dq, adq b ; q)n ( c , d ; q)n 173 − c d b ( ad , bc ; p)m+1 ( d1 , ad ; q)m+1 bc 1 1 ( 1c , ad 2 ; q)m+1 ( a , b ; p)m+1 ) . 174 M.V. SUBBARAO AND A. VERMA Jain and Verma [6] used transformations of q-hypergeometric series to obtain a summation formula involving three independent bases: (1.4) pP n X (β; p)k (c; q)k (y; p)k ( βyc d2 ; q )k [(1 − βcy k k y k −k )(1 d p P )(1 − d P q βy pP pP cyP (dq; q)k ( βcp d ; p)k ( d q ; q )k ( d2 ; P )k k=−m (1 − β)(1 − c)(1 − y)(1 − = (c − d) βcy d2 ) ( − βd pk q −k )]q k d d d pP ( βc ; p)m+1 ( d1 ; q)m+1 ( cy ; P )m+1 ( βy ; q )m+1 2 d ; pP ( β1 ; p)m+1 ( 1c ; q)m+1 ( y1 ; P )m+1 ( βcy q )m+1 ) pP pP (βp; p)n (cq; q)n (yP ; P )n ( βcy d2 q ; q )n − βcp , βy pP pP ( d ; p)n (dq; q)n ( cyP d ; P )n ( d q ; q )n which for P = q reduces to the Gasper-Rahman’s summation formula (1.3). The proof of (1.4) could be given by considering βk = pP pP q ; q )k cy βy pP pP ( βc d p; p)k (dq; q)k ( d P ; P )k ( d q ; q )k (βp; p)k (cq; q)k (yP ; P )k ( βcy d2 and observing that (1.5) βyc pP pP ; d2 q q ∆βk = (βp; p)k−1 (cq; q)k−1 (yP ; P )k−1 k−1 h i βcy k k β k −k y k −k (1 − d2 p P )(1 − d P q )(1 − d p q ) × , βy pP pP cyP ( βcp d ; p)k (dq; q)k ( d q ; q )k ( d ; P )k and summing for k from −m to n (m, n are non-negative integers) and using the usual convention: m≤n n Am Am+1 · · · An Y (1.6) Ak = 1 m=n−1 −1 k=m (An+1 An+2 · · · Am−1 ) m ≥ n − 2. Chu [2] obtained a generalization of Gasper-Rahman’s formula (after renaming suitably the sequences so as to remove redundant sequences) (1.7) n X k=−m k−1 Q (1 − αak bk )(bk − (1 − αa0 b0 )(b0 − ak αd ) a0 dα ) [(1 − aj )(1 − j=0 k Q = α(1 − − cbj )(1 − [(1 − αbj )(1 − αdbj )(1 − j=1 a0 α2 d d )(1 − b0 c)(1 − c b0 ) a0 αa0 b0 )(1 − αc )(b0 − dα )(1 − αd c ) (1 − a0 )(1 − aj d )(1 αaj c )(1 α2 d c bj )] − c dα aj )] SOME SUMMATIONS OF q-SERIES BY TELESCOPING ( × n Y − 2 (1 − aj )(1 − dj )(1 − bj c)(1 − αc d bj ) αa c (1 − αbj )(1 − αbj d)(1 − c j )(1 − αd aj ) " (1 − αbj )(1 − αdbj )(1 − j=1 0 Y a " (1 − aj )(1 − j=−m aj d )(1 αaj c )(1 − cbj )(1 − c αd aj ) α2 d c bj ) − 175 # #) where haj i and hbj i are arbitrary sequences such that none of the terms in the denominators vanish. This reduces to the Gasper-Rahman summation formula on setting ak = apk , bk = q k and replacing α and d by d and a/b, respectively. In this paper we obtain in §2 a generalization of Chu’s summation formula (1.7) involving four arbitrary sequences, which on specialization yields an extension of (1.1) to a summation formula with four independent bases p, q, P and Q and incorporating (1.7) as a special case. An expansion of the ∞ P series An Bn (wx)n into a series involving three independent bases is den=0 veloped. A transformation of a series involving eight independent sequences is also developed. The note is concluded by obtaining in §3 some summation formulas which are different from the known ones by telescoping of series including q-Paff-Saalschütz’s summation formula for a terminating balanced 3 φ2 [q]. §2. We begin this section by proving the summation formula: If huk i, hvk i, hwk i and hzk i are arbitrary sequences such that none of the terms in the denominators vanish and M, N are non-negative integers then (2.1) n (1 − uk vk wk zk )(1 − X k=−m (1 − u0 v0 w0 z0 )(1 − k−1 Q × j=0 k Q j=1 = w k zk uk vk )(1 − w 0 z0 u0 v0 )(1 − v k zk uk wj )(1 − v 0 z0 u0 w0 )(1 − uk zk vk w k ) u0 z0 v0 w0 ) [(1 − u2j )(1 − vj2 )(1 − wj2 )(1 − zj2 )]( uk vzkkwk ) [(1 − v j w j zj uj )(1 − uj w j zj vj )(1 − uj v j zj wj )(1 − uj v j w j zj )] (1 − u20 )(1 − v02 )(1 − w02 )(1 − z02 ) v 0 z0 u0 z0 0 z0 (1 − u0 v0 w0 z0 )(1 − w u0 v0 )(1 − u0 w0 )(1 − v0 w0 ) ( n " # Y (1 − u2j )(1 − vj2 )(1 − wj2 )(1 − zj2 ) × v w z u w z u v z u v w (1 − j ujj j )(1 − j vjj j )(1 − jwjj j )(1 − j zjj j ) j=1 176 M.V. SUBBARAO AND A. VERMA j=−m v j w j zj uj w j zj uj v j z j uj v j w j uj )(1 − vj )(1 − wj )(1 − zj ) (1 − u2j )(1 − vj2 )(1 − wj2 )(1 − zj2 ) k Y (1 − u2j )(1 − vj2 )(1 − wj2 )(1 − zj2 ) 0 Y − " (1 − #) . Proof. Let τk = j=1 " (1 − v j w j zj uj )(1 − uj w j zj vj )(1 − uj v j zj wj )(1 − # uj v j w j zj ) . Then by straight forward calculations, we get i k−1 Q h (1 − u2j )(1 − vj2 )(1 − wj2 )(1 − zj2 ) ∆τk = j=1 k Q j=1 h (1 − v j w j zj uj )(1 − uj w j zj vj )(1 − uj v j zj wj )(1 − i uj v j w j zj ) ( uk vzkkwk ) wk zk vk zk u k zk × (1 − uk vk wk zk ) 1 − 1− 1− uk vk uk wk vk wk where ∆τk = τk − τk−1 . Now summing with respect to k from −m to n, and n P using the fact that τk = τn − τ−m−1 and keeping in mind (1.6), we get k=−m (2.1) on simplification. √ pa j √ √ By setting uj = a pj , vj = c q j , wj = b P j , zj = d bc Q in (2.1), we get a summation formula involving four independent bases: (2.2) n X (1 − adpk q k P k Qk )(c − k=−m dP k Qk )(1 pk q k pk Qk 2k bpk P k )(1 − ad bc q k P k )q dq k Qk b ad d )(1 − bc ) − (1 − ad)(c − d)(1 − 2 × = 2 (a; p2 )k (c; q 2 )k (b; P 2 )k ( ad bc ; Q )k (d qPpQ ; qPpQ )k ( ad c pP Q pP Q ad pqQ pqQ bc pqP pqP q ; q )k ( b P ; P )k ( d Q ; Q )k ad2 bc ) − ad bc ) (1 − a)(1 − b)(1 − c)(1 − (1 − ad)(c − d)(1 − db )(1 ( 2 2 2 (ap2 ; p2 )n (cq 2 ; q 2 )n (bP 2 ; P 2 )n ( ad bc Q ; Q )n × pP Q pP Q ad pqQ pqQ bc pqP pqP ( dqPp Q ; qPpQ )n ( ad c q ; q )n ( b P ; P )n ( d Q ; Q )n ) c pP Q b pqQ d pqP ( d1 ; qPpQ )m+1 ( ad ; q )m+1 ( ad ; P )m+1 ( bc ; Q )m+1 − . bc ( a1 ; p2 )m+1 ( 1c ; q 2 )m+1 ( 1b ; P 2 )m+1 ( ad ; Q2 )m+1 2 2 2 Summation formula (2.2), on setting Q = pP q and replacing p , q , P , a, b, d by p, q, P, β, y, cy d , respectively, reduces to the summation formula (1.4), which in turn incorporates (1.3) and (1.2) as special cases. SOME SUMMATIONS OF q-SERIES BY TELESCOPING 177 It may be pointed out that (2.2) √ reduces topChu’s summation formula p p √ √ (1.7) on setting uj = aj , vj = aj / d, wj = cbj , zj = α d/c bj . u0 Setting m = 0 in (2.2), replacing zi by αzi and setting α = v0 w , we get 0 z0 that (2.3) wj zj v z v0 w0 z0 0 z0 − ujjwjj )( v0 w uj vj )( u0 u0 − v02 )(1 − w02 )(v02 w02 − u20 ) n ( v0 w0 z0 − u v w z )( v0 w0 z0 − X j j j j u0 u0 (1 − u20 )(1 j=0 j−1 Q i=0 × k Q i=1 0 z0 [( v0 w − u0 [(1 − u2i )(1 − vi2 )(1 − wi2 )( vi wi zi v0 w0 z0 ui )( u0 z03 v0 w0 u30 n Y v wz × ( o u00 0 − i=1 − v02 w02 z02 u20 ui wi zi v0 w0 z0 vi )( u0 − − uj zj uj vj wj vj wj ) zj − zi2 )] ui vi zi u0 wi )( v0 w0 z0 − ui vi w i zi )] = v02 w02 z02 − zi2 ) u20 ui wi zi v0 w0 z0 u0 − uiwviizi )( v0 w vi )( u0 0 z0 (1 − u2i )(1 − vi2 )(1 − wi2 )( vi wi zi v0 w0 z0 ui )( u0 − − ui vi w i zi ) . Next, using (2.3), the following transformation involving eight arbitrary sequences is obtained: (2.4) n n−k X Y 2 4 k=0 i=1 × (1 − Ui2 )(1 − Vi2 )(1 − Wi2 )( 0 Z0 − ( V0 W U0 0 z0 0 z0 − uk vk wk zk )( v0 w − ( v0 w u0 u0 (1 − u20 )(1 k−1 Q × × k Q 0 z0 [( v0 w − u0 Ui Wi Zi 0 Z0 )( V0 W Vi U0 0 z0 − ( v0 w u0 − 2 2 2 v0 w 0 z0 u2 0 u i w i zi 0 z0 )( v0 w vi u0 Ui Vi Zi U0 )( V0 W Wi 0 Z0 − − − vi wi zi 0 z0 )( v0 w ui u0 − − zi2 )] ui vi zi u0 )( v0 w wi 0 z0 − 2 2 2 v0 w 0 z0 − zi2 ) u2 0 u i w i zi u0 0 z0 )( v0 w − uiwviizi )( v0 w vi u0 0 z0 Wj Zj Uj Vj (1 − U02 )(1 − V02 )(1 − Ui Vi Wi ) Zi uk zk uk vk wk ) zk vk wk ui vi wi )] zi (1 − u2i )(1 − vi2 )(1 − wi2 )( n ( V0 W0 Z0 − U V W Z )( V0 W0 Z0 − X j j j j U0 U0 j=0 − 3 − Zi2 ) w k zk zk 0 z0 0 z0 )( v0 w − uvkkw )( v0 w uk vk u0 u0 k − v02 )(1 − w02 )(v02 w02 − u20 ) [(1 − u2i )(1 − vi2 )(1 − wi2 )( vi wi zi 0 z0 )( v0 w ui u0 V0 W0 U03 z03 3 Z0 v0 w0 u30 " n Y i=1 × − i=0 i=1 = Vi Wi Zi 0 Z0 )( V0 W Ui U0 V02 W02 Z02 U02 # − ui vi wi ) zi Vj Zj 0 Z0 )( V0 W Uj Wj U0 W02 )(V02 W02 − U02 ) 0 Z0 )( V0 W − U0 − Uj Zj Vj Wj ) 5 178 M.V. SUBBARAO AND A. VERMA × Uj Vj Wj Zj j−1» × Y (1 − Ui2 )(1 − Vi2 )(1 − Wi2 ) V02 W02 Z02 − Zi2 U02 „ i=0 „ × × v0 w0 z0 vn−i wn−i zn−i − u0 un−i «„ « v0 w0 z0 un−i wn−i zn−i − u0 vn−i «– 1 j h“ Q V0 W0 Z0 U0 i=0 j−1 Q h“ i=0 × j h Q − Ui Wi Zi Ui v0 w0 z0 u0 − ”“ V0 W0 Z0 U0 un−i vn−i zn−i wn−i − Ui Wi Zi Vi ”“ u0 v0 w0 z0 ”“ V0 W0 Z0 U0 − un−i vn−i wn−i zn−i 2 2 (1 − u21+n−i )(1 − v1+n−i )(1 − w1+n−i ) i=0 “ 2 w2 z 2 v0 0 0 u2 0 − Ui Vi Zi Wi ”“ U0 V0 W0 Z0 − Ui Vi Wi Zi ”i ”i 2 − z1+n−i ”i . Transformation (2.4) can be proved by expanding the first product on the left hand side by using (2.3) (with n replaced by n − k), interchanging the order of summations and evaluating the inner sum by using (2.3) once again and simplifying to get the right hand side of (2.4). √ Transformation (2.4), √ on replacing ui , vi , wi , zi , Ui , Vi , Wi , Zi by a pi , √ √ i √ i i √ e i , respectively, reduces on some c q , b P , Q , A pei , C qei , B Pei , Q simplification to the following transformation of q-series involving eight independent bases: (2.5) n (1 − apk q k P k Qk )( q X pk k k P − cq 2k )(1 − b pqk Q k )(1 − (1 − a)(1 − c)(1 − b)(1 − k=0 × k P k Qk a pk Qk bc q k P k ) a bc ) a (a; p2 )k (c; q 2 )k (b; P 2 )k ( bc ; Q2 )k ( qPpQ ; qPpQ )k ( ac pP Q pP Q pqP pqP a pqQ pqQ q ; q )k ( b P ; P )k (bc Q ; Q )k −n P e−n Q e −n × ( qe × ee e e −n pePeQ e −n peqeQ e pe−n Pe−n Q pe−n qe−n Q ; qePpeQ )k ( C ; qe )k ( B ; e )k A A pe−n qe−n Pe−n P −2n −2n e−2n e −2n ; Q e 2 )k ( pe A ; pe2 )k ( qe C ; qe2 )k ( P B ; Pe2 )k ( BC A Q pe−n qe−n Pe−n peqePe ; e −n e BC Q Q ! k 2 = 2 (ap2 ; p2 )n (cq 2 ; q 2 )n (bP 2 ; P 2 )n ( aQ bc ; Q )n ( qPpQ ; qPpQ )n ( ac pP Q pP Q pqP pqP a pqQ pqQ q ; q )n ( b P ; P )n (bc Q ; Q )n SOME SUMMATIONS OF q-SERIES BY TELESCOPING 179 e e pePeQ e e peqeQ peqePe peqePe pePeQ A peqeQ e ; Q e )n qe ; qe )n ( B Pe ; Pe )n (BC Q e2 Q e 2 )n (ae p 2 ; pe2 )n (C qe2 ; qe2 )n (B Pe2 ; Pe2 )n ( ABC ;Q A ( qePpeQ ; qePpeQ )n ( C ee × ee j ej ej B pej Pej e j )(1 qej Q A B)(1 − BC ) e j )( qe P j Q − C qe2j )(1 − n (1 − Ae pj qej Pej Q X pe × (1 − A)(1 − C)(1 − j=0 − ej A pej Q BC qej Pej ) A e2 (A; pe2 )j (C; qe2 )j (B; Pe2 )j ( BC ; Q )j × A ( qePpeQ ; qePpeQ )j ( C ee (q × −n P −n Q−n e e pePeQ e e peqeQ peqePe peqePe pePeQ A peqeQ e ; Q e )j qe ; qe )j ( B Pe ; Pe )j (BC Q −n −n −n p−n P −n Q−n pP Q ; q )j ( ab p Pq −nQ ; pqQ P )j q −n −2n −2n −2n −2n ( p a ; p2 )j ( q c ; q 2 )j ( P b ; P 2 )j ( bcQa ; Q2 )j p−n × ee ; qPPQ )j ( ac p−n q −n P −n pqP ; bcQ−n Q . j Transformation (2.5) is a generalization of Gasper’s [4, Ex. 3.21] quadbasic transformation (from which Gasper deduces a transformation of a halfpoised 10 φ9 into another half-poised 10 φ9 [4, Ex. 3.24]) to which it reduces e = qe and then replacing p2 , q 2 , pe2 and on setting Q = q, P = p, Pe = pe, Q 2 qe by p, q, P and Q, respectively. ∞ P Next, we obtain an expansion of Ar Br (xw)r /(q; q)r terms of q-series r=0 having three independent bases. In the summation formula (2.2), setting m = 0, c = q −2n , where n is a non-negative integer and letting d → 1, we get n (1 − apj q j P j Qj )( q X j P j Qj pj − q 2j−2n )(1 − (1 − a)(1 − q −2n )(1 − b)(1 j=0 × bpj P j pj Qj )(1 − ab qj−2n ) q j Qj Pj a 2n −bq ) (a; p2 )j (q −2n ; q 2 )j (b; P 2 )j ( ab q 2n ; Q2 )j Q pP Q a ( qPpQ ; qPpQ )j ( qapP 1−2n ; q )j ( b pqQ pqQ bpq 1−2n P pqP ; Q )j P ; P )j ( Q = δn,0 . Replacing a and b by apr q r P r Qr and bpr q −r P r Q−r respectively, where r is 2 2 2 a non-negative integer, setting Q = pq P and then replacing p , q , P , n by p, q, P, m, respectively, we get (2.6) δm,0 = m X (1 − apj+r q j+r )(1 − bP j+r q −j−r )(1 − ab pr+j q 2r+m P −r−j )q j (1 − apr q r )(1 − bP r q −r )(1 − ab pr q 2r+m P −r ) j=0 × (apr q r ; p)j (q −m ; q)j (bP r q −r ; P )j ( ab pr q 2r+m P −r ; pq P )j . a r+1 1+2r −r−1 pq 1+r r+m −m−r (q; q)j (ap q ; p)j ( b p q P ; P )j (bq P r+1 ; P )j 180 M.V. SUBBARAO AND A. VERMA But, we know that Br xr = ∞ X (1 − a b m=0 r −r pr+m q 2r+2m P −r−m )( ab pr q 2r P −r ; pq P )m (apq ; p)r (bP q ; P )r (1 − ab pr q 2r P −r )(q; q)m (apq r+m ; p)r (bP q −m−r ; P )r × {q −rm Br+m Cr,m xr+m δm,0 }, where hBr i and hCr,m i are arbitrary sequences of complex numbers such that Cr,0 = 1 for r = 0, 1, 2, . . . Substituting for δm,0 from (2.6), interchanging the order of summation and setting j = n − r and m = n + k − r, we get Br xr = ∞ X ∞ X (−)n (1 − apn q n )(apq r ; p)n−1 (1 − bP n q −n )(bP q −r ; P )n−1 k=0 n=r (1 − ab pn+k q 2n+2k P −n−k )xn+k (q −n ; q)r × (apq n+k ; p)n (bP q −n−k ; P )n (q; q)n (q; q)k a n n+k+r −n a pn+1 q n+r+1 pq 1− p q P ; b b P n+1 P k−1 2 r+1 n+k+r+1 −r−1 pq n(r−k−n)+ n2 + n 2 × p q P ; Bn+k Cr,n+k−r q . b P n−r−1 a r Ar w and summing from r = 0 to ∞ and interMultiplying both sides by (q;q) r changing the order of summation on the right hand side, we get ∞ X Ar Br r=0 (xw)r (q; q)r n ∞ X (1 − apn q n )(1 − bP n q −n )(−x)n q ( 2 )+n = (q; q)n n=0 × ∞ X k=0 n+k 2n+2k q (1 − ab p P n+k )xk Bn+k (q; q)k (apq n+k ; p)n (bP q −n−k ; P )n ∞ X (q −n ; q)r (apq r ; p)n−1 (bP q −r ; P )n−1 × (q; q)r r=0 a pn q n+k+r a pn+1 q n+r+1 pq × 1− ; b Pr b P n+1 P k−1 a pr+1 q n+k+r+1 pq × ; Cr,n+k−r wn q n(r−k−n) Ar , b P r+1 P n−r−1 which is a generalization of Gasper’s bibasic expansion formula [4, (3.7.6)] to which it reduces on setting P = p. It may be noted that on setting P = p the SOME SUMMATIONS OF q-SERIES BY TELESCOPING 181 terms in {· · · } of the above expression combine to yield ( ab q n+r+1 ; q)n+k−r−1 as in the bibasic expansion formula of Gasper [4, (3.7.6)]. §3. All the summation formulae proved so far are for one generalization of (1.1), a very-well-poised q-series. We next derive a summation formula which gives the sum of a balanced series. Let hxi i, hyi i and hzi i be arbitrary sequences and a an indeterminate so that none of the terms in the denominators vanish and m, n are non-negative integers. Then yk xk )(1 − az )zk (1 − az k k y0 x0 (1 − az0 )(1 − az0 ) k=−m n X k−1 Q [(1 − xi )(1 − yi )] i=0 k Q [(1 − azi )(1 − i=1 (3.1) xi y i azi )] (1 − x0 )(1 − y0 ) = y0 x0 a(1 − az )(1 − az ) 0 ( n "0 # " #) 0 i yi Y Y ) (1 − azi )(1 − xaz (1 − xi )(1 − yi ) i × − . i yi (1 − azi )(1 − xaz ) (1 − x )(1 − y ) i i i i=1 i=−m For proving (3.1) we consider " # k Y (1 − xi )(1 − yi ) τk = . i yi (1 − azi )(1 − xaz ) i i=1 Then by straight forward calculations we find that ∆τk = azk k−1 Q [(1 − xi )(1 − yi )] xk yk i=1 1− 1− , k azk azk Q [(1 − azi )(1 − xzi yi i )] i=1 which on summing over k from −m to n, gives (3.1) after using (1.6). In view of this it is natural to look for a telescoping proof of the q-Paff-Salschütz summation formula [4, (1.7.2)] " # a, b, q −n ; q, q ( ac , cb ; q)n ab 1−n (3.2) Sn ≡ 3 φ2 = . c c, c q (c, ab ; q)n To this end define for non-negative integers n and r F (n, r) = (a, b, q −n ; q)r q r 1−n ; q) (q, c, ab r c q 182 M.V. SUBBARAO AND A. VERMA and G(n, r) = (a, b; q)r+1 (q −n ; q)r . 1−n ; q) (q, c, ab r c q Notice that S0 = 1, G(n, n + 1) = G(0, −1) = 0. By straightforward calculations we can verify that c n c n cq n n F (n + 1, r) 1− q 1 − q F (n, r) − (1 − cq ) 1 − a b ab c n − q [G(n, r) − G(n, r − 1)] = 0. ab Summing over r from 0 to n + 1 and using G(n, n + 1) = 0 = G(0, −1), we get c c c 1 − q n 1 − q n Sn = (1 − cq n ) 1 − Sn+1 a a ab which yields ( c , c ; q)n S0 . Sn = a cb (c, ab ; q)n By using S0 = 1, we get (3.2). Acknowledgment. The authors wish to thank the referee for helpful comments. References [1] R.P. Agarwal, On the partial sums of series of hypergeometric type, Proc. Camb. Phil. Soc., 49 (1953), 441-445. [2] W. Chu, Inversion techniques and combinatorial identities, Boll. U.M.I., 7(7) (1993), 737-760. [3] G. Gasper, Summation, transformation, and expansion formulas for bibasic series, Trans. Amer. Math. Soc., 312 (1989), 257-277. [4] G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, 1990. [5] , An indefinite bibasic summation formula and some quadratic, cubic and quartic summation and transformation formulas, Canadian J. Math., 42 (1990), 1-27. [6] V.K. Jain and A. Verma, Summation formulae for bibasic and polybasic hypergeometric series, J. Indian Math. Soc., 60 (1996). Received January 2, 1998 and revised April 29, 1998. The first author was partially supported by an NSERC grant. University of Alberta Edmonton, Alberta T6G 2G1 Canada E-mail address: [email protected] University of Roorkee Roorkee 247667, U.P. India SOME SUMMATIONS OF q-SERIES BY TELESCOPING E-mail address: [email protected] 183 PACIFIC JOURNAL OF MATHEMATICS Vol. 191, No. 1, 1999 DEHORNOY’S ORDERING OF THE BRAID GROUPS EXTENDS THE SUBWORD ORDERING Bert Wiest We give a simple proof of the result of Laver and Burckel that inserting a conjugate of a positive standard generator of the braid group anywhere in a given braid yields a braid which is larger in the sense of Dehornoy. P. Dehornoy has defined a right invariant total ordering of the braid group Bn for all n ∈ N (see [3], [4]), and this was reinterpreted in [6] in more geometrical terms. Here we are using this interpretation to give a quick proof of the following: Theorem 1. For any braids α, β1 , β2 ∈ Bn and any i ∈ {1, . . . , n − 1} we have β1 (ασi α−1 )β2 > β1 β2 . Since the ordering is right-invariant this is equivalent to the following, with γ = β1 α: Theorem 10 . γσi > γ. For any braid γ ∈ Bn and any i ∈ {1, . . . , n − 1} we have It follows that Dehornoy’s ordering extends the (partial) subword ordering defined in [5]. Theorem 1 was first proved by Laver [8] and Burckel [2] using very different methods. As explained in [8], it can be combined with a theorem of Higman [7] to prove that the restriction of the ordering to the positive braid monoid Bn+ is a well-ordering. We briefly recall the definition of the ordering of Bn given in [6]. Let D2 be the unit disk in C, and let Dn be equal to D2 with n holes in the real line, labelled 1 to n. The holes divide the real interval [−1, 1] into n + 1 line segments which we label 1 to n + 1. Now any braid γ determines a way of sliding the holes about in D2 . Extending this movement to an isotopy of D2 which is fixed on ∂D2 , we obtain, at the end of the isotopy, a self homeomorphism of Dn ; this self homeomorphism maps the n + 1 line segments to n + 1 disjoint simple curves, again numbered 1 to n + 1, and the image of the whole interval [−1, 1] under the self homeomorphism is called a curve diagram. If Γ is a curve-diagram in Dn of some braid γ, and ∆ is another curve diagram of some braid δ, then we can reduce Γ and ∆ with respect to each other, i.e. we can ‘pull the diagrams tight’. Then the braid γ is called j-larger than δ if the curves number 1,. . . ,j − 1 of Γ and ∆ coincide and 183 184 BERT WIEST an initial segment of the jth curve of Γ lies in the upper component of Dn − ∆. It is proved in [6] that this is equivalent to the braid γδ −1 being representable by a word w1 σj w2 . . . wl−1 σj wl , where w1 , . . . , wl are words in ±1 ±1 the letters σj+1 , . . . , σn−1 . If γ is j-larger than δ for some j, then we say γ is larger than δ. If the curves number 1,. . . ,j − 1 of Γ and ∆ coincide (with no further restrictions on the subsequent curves), then we say γ and δ are j − 1-parallel (j ∈ {1, . . . , n}); equivalently, γδ −1 can be represented by a ±1 word not containing the letters σ1±1 ,. . . ,σj−1 . The proof of Theorem 10 is in two steps. We shall deduce Theorem 10 from the following, seemingly weaker, result: Proposition 2. For any braid γ ∈ Bn and any i ∈ {1, . . . n − 1} the braid γσi is not 1-smaller than the braid γ; equivalently, γσi is either 1-larger than, or 1-parallel to γ. Proof of Proposition 2. We consider the intersections of a curve diagram Γ with the vertical lines indicated in Figure 1, which divide Dn into n + 1 regions, labelled 1 to n + 1. We say Γ is v-reduced if there are no disks in Dn bounded by precisely one segment of some curve of Γ and one segment of vertical line. By a sequence of isotopies across such disks we can turn Γ into a v-reduced diagram. 1 1 2 3 4 ·σ3 1 2 3 4 5 Figure 1. Vertical lines dividing Dn into a number of regions. We say an isotopy of Γ is a v-equivalence, if it leaves Γ transverse to the vertical lines at all times. In particular, a v-equivalence does not change the number of intersections of Γ with the vertical lines. We now give a recipe how to obtain a v-reduced curve diagram for the braid γσi from a v-reduced curve diagram Γ for the braid γ. The crucial observation is that under right multiplication by σi a v-reduced curve diagram changes only in the regions labelled i, i + 1, and i + 2, according to a simple set of rules. We call the union of these three regions the augmented i + 1-region. We equip all curves of Γ consistently with an orientation such that, when stuck together, they form an oriented curve in D2 starting at −1, through all holes of Dn , ending at 1. For every curve of Γ we consider the segments in ORDERINGS OF THE BRAID GROUP 185 which it intersects the augmented i-region. If the whole curve is contained in the i + 1st region itself, connecting the ith and the i + 1st hole, then right multiplication by σi has simply the effect of reversing the orientation of the curve (see Figure 2(a)). If a segment of curve can, by a v-equivalence, be made disjoint from the straight line from the ith to the i + 1st hole, then it is unaffected by the right multiplication by σi (Figure 2(b)). (a) i i+1 . ·σi i (b) . ·σi i Figure 2. Unaffected by performing σi . If a segment of curve cannot be made disjoint from the straight line from the ith to the i + 1st hole, then it is affected by the right multiplication by σi . We shall find that there are ten Z-families of possibilities for what such a segment of curve can look like. If both ends of the segment of curve lie in the i − 1st vertical line (i.e. the left boundary of the augmented i + 1-region), then it is easy to check that in a neighbourhood of the i + 1st region it looks like one of the curves in Figure 3, and under right multiplication by σi the diagram changes as indicated. . σi . σi . σi . σi . σi . σi . σi . σi . σi . σi . σi . σi Figure 3. Two Z-families of possibilities ((1) and (2)). Note that all curves in Figure 3 can have two different orientations so Figure 3 represents two different Z-families of curve segments. Also note that we can assume that the right multiplication by σi leaves the curve diagram fixed in a neighbourhood of the boundary of the augmented i + 1region, and Figure 3 shows its effect only ‘well inside’ this region. If both ends of the segment of curve lie in the i + 2nd vertical line (i.e. the right boundary of the augmented i + 1-region), then we have another two Z-families of possibilities ((3) and (4)). Figure 3, turned by 180◦ , illustrates these two families. In the case that the two ends of the segment of curve lie on opposite sides of the augmented i + 1-region (i.e. one in the i − 1st and one in the i + 2nd vertical line), we obtain two more Z-families of possibilities ((5) and (6)), which differ only in the orientation of the curve segment (Figure 4). Again, it is easy to see that in this case these are the only possibilities. 186 BERT WIEST .·σi i .·σi i .·σi i .·σi i .·σi i Figure 4. Two Z-families of possibilities ((5) and (6)). Finally, for the case that one of the ends of a curve segment lies in the boundary of the augmented i-region and the other end is in a hole of Dn , we have four more Z-families: The one in Figure 5 (Family (7)), the one represented by Figure 5 turned by 180◦ (Family (8)), and the same two families with the orientation of the curve segments reversed (Families (9) and (10)). . ·σ ii . ·σ ii . ·σ ii . ·σ ii . ·σ ii ·σi i Figure 5. One more Z-family of possibilities (7). This completes our construction of a v-reduced curve diagram for γσi from the diagram Γ. Our next aim is to compare the v-reduced curve diagrams of γ and γσi , in order to decide which one is larger. The proof of the following lemma is similar to the proof in [6] that three curve diagrams can be reduced with respect to each other. Lemma 3. If Γ and ∆ are two v-reduced curve diagrams, then ∆ can be reduced with respect to Γ by an isotopy which is a v-equivalence. Given a v-reduced curve diagram Γ, we use the above recipe to determine a v-reduced curve diagram ∆ of γσi . Then we reduce ∆ with respect to Γ to obtain a curve diagram ∆0 , and Lemma 3 tells us that this can be done by an isotopy of ∆ which is a v-equivalence; that means, the intersections of ∆ with the vertical lines can slide up and down without crossing the holes of Dn , and no intersections are created or cancelled in the course of this isotopy. There are now two possibilites: (1) γ and γσi are 1-parallel, i.e. the first curves Γ1 and ∆01 of Γ and ∆0 coincide. (2) Γ1 and ∆01 do not coincide. In case (1) we are done. In case (2) we walk along the curve Γ1 , starting at −1, reading off a finite sequence of numbers according to Figure 6. The first number lies in {1, 2, 3}, the subsequent ones in {1, . . . , 4}. The sequence ends with a 2 or a 3. We do the same for ∆01 , and obtain a different sequence ORDERINGS OF THE BRAID GROUP 187 of numbers (for if the sequences were the same, then the curves Γ1 and ∆01 would coincide). Since the reduction of ∆ with respect to Γ was a vequivalence, the number sequences obtained by reading along ∆1 and ∆01 agree. (a) 3 (b) 4 3 2 1 1 2 (c) 4 3 2 1 (d) 1 2 3 (e) 2 4 1 3 4 Figure 6. Reading a sequence of numbers off the curves Γ1 and ∆01 . Now comes the key step of the whole proof. By carefully checking through all ten Z-families described above it can be seen that the number sequence associated with ∆1 (obtained from Γ1 by applying σi ) is always lexicographically larger than the number sequence of Γ1 ; i.e. the first k − 1 terms of the two sequences agree (k ∈ N), but the kth number read off ∆1 is larger than the kth number read off Γ1 . For instance, from the two leftmost diagrams in Figure 4 one reads . . . 114314 . . . and . . . 13 . . . respectively (independently of orientation), and the second sequence is lexicographically larger. Similarly, from the diagrams in Figure 5 one reads . . . 1142, . . . 12, . . . 2, . . . 33, and . . . 3413, which is lexicographically increasing. The other cases are similar and left to the reader. Consider now the subarcs of the curves Γ1 and ∆01 consisting of the first k curve segments as in Figure 6. Since Γ and ∆01 are reduced with respect to each other, these arcs do not intersect. It follows that an initial segment of ∆01 lies in the upper component of Dn − Γ, ie that γσi is 1-larger than γ. This completes the proof of Proposition 2. Proof of Theorem 10 . To see that Proposition 2 implies Theorem 10 we assume, for a contradiction, that there exists a braid γ and an i ∈ {1, . . . , n} such that γσi γ −1 is j-negative with j > 1. Then γσi γ −1 can be represented ±1 by a word not containing the letters σ1±1 ,. . . ,σj−1 , and only negative powers of σj . We consider the natural epimorphism π : Bn → Sn , from the braid group to the symmetric group. We have π(σi ) = (i, i + 1), π(γ −1 ) = (π(γ))−1 , and π(γσi γ −1 )(k) = k for k = 1, . . . , j − 1. It follows that π(γ)(k) ∈ / {i, i + 1} for k = 1, . . . , j − 1. Therefore the braid σ e on n − j + 1 strings which is obtained from the braid σi by removing the strings number π(γ)(1), . . . , π(γ)(j − 1) is again a positive standard generator of Bn−j+1 . Similarly, by removing from the braid γ the strings starting in the j − 1 leftmost positions we obtain a braid γ e ∈ Bn−j+1 . 188 BERT WIEST Then the braid γ eσ eγ e−1 ∈ Bn−j+1 is a conjugate of the positive standard generator σ e. On the other hand, the braid γ eσ eγ e−1 is obtained from γσi γ −1 by removing the j − 1 leftmost strings. Therefore it can be represented by a word containing the letter σ1−1 , but not σ1 , thus contradicting Proposition 2. Added in proof. Consider the monoid Π = {π ∈ Bn : βπ > β ∀ β ∈ Bn } which is closed under conjugation by elements of Bn . I do not know a complete set of generators of Π. S. Orevkov has pointed out that, in addition to positive standard generators, Π also contains all braids of the form ασ1 σ2 . . . σn−1 σn−1 . . . σ1 , where α is any braid not containing σ1±1 . Acknowledgements. Thanks to P. Dehornoy and C. Rourke for helpful comments. References [1] J. Birman, Braids, links, and mapping class groups, Annals of Math. Studies, 82, Princeton University Press, Princeton, 1975. [2] S. Burckel, The wellordering on positive braids, J. Pure Appl. Algebra, 120(1) (1997), 1-18. [3] P. Dehornoy, Braid groups and left distributive operations, Trans. A.M.S., 343 (1994), 115-150. [4] , A fast method of comparing braids, Adv. in Math., 125 (1997), 200-235. [5] E. Elrifai and H. Morton, Algorithms for positive braids, Quart. J. Math. Oxford, 45 (1994), 479-497. [6] R. Fenn, M.T. Greene, D. Rolfsen, C. Rourke and B. Wiest, Ordering the braid groups, Pac. J. Math., 191(1) (1999), 49-74. [7] G. Higman, Ordering and divisibility in abstract algebras, Proc. London Math. Soc., 2 (1952), 326-336. [8] R. Laver, Braid group actions on left distributive structures and well-orderings in the braid group, J. Pure Appl. Algebra, 108(1) (1996), 81-98. Received February 17, 1998. The author was supported by a Marie Curie research training grant from the European Community. Université de Provence 39 Rue Joliot-Curie 13453 Marseille cedex 13 France E-mail address: [email protected] PACIFIC JOURNAL OF MATHEMATICS Vol. 191, No. 1, 1999 A UNIFIED METHOD OF CONSTRUCTION OF ELLIPTIC CURVES WITH HIGH MORDELL–WEIL RANK Hizuru Yamagishi By using the twist theory, we reduce the problem of constructing elliptic curves of rank n (n ≥ 1) with generators to the problem of finding rational points on a certain variety Vn . By parametrizing all rational points on Vn (1 ≤ n ≤ 7), we get all elliptic curves of at least rank n (n ≤ 7). 1. Introduction. The purpose of this paper is to describe a unified method of construction of elliptic curves with given Mordell-Weil rank, and to show it is powerful enough to produce every known example in principle. In view of the fact that there is no general algorithm to give an elliptic curve with high rank, our method might shed a light on this area of active research. The main ingredient in this paper is the twist theory as is developed in [3]. One of our main theorems says that the twists give us every elliptic curve with high Mordell-Weil rank. Once this is established, we naturally come to consider a variety Vn parametrizing a family of elliptic curves with a given rank n, and we show that almost every rational point on this variety gives an elliptic curve with rank ≥ n. Thus our method should provide us with every known example as a rational point on it. We will show that this is indeed the case. In this paper, we parametrize the rational points on Vn in each case of n = 1 to 7. As is mentioned above parametrizing the rational points on Vn is equivalent to getting every elliptic curve of rank at least n (n ≤ 7). Moreover, the variety Vn is expected to be useful for solving other problems. For example, in [5] it is used to construct a family of elliptic curves of rank 2 with given j-invariant. Moreover in [6], it is used to give a family of elliptic curves of rank 6 with a nontrivial rational two-torsion point. The point which is worthy of special mention is that we get very easily the equation of any elliptic curve which corresponds to a rational point on Vn . 189 190 HIZURU YAMAGISHI The present paper is organized as follows. In Section 2, we show every elliptic curve with generators comes from a twist. Then for each rank n we define the variety Vn mentioned above, which plays a very important role throughout the paper, and construct a generic elliptic curve with its generators as a generic fiber of a certain family of elliptic curves parametrized by Vn . Furthermore we show that we can get a given elliptic curve by specializing this family at a certain rational point on Vn . In Section 3, we focus our attention on the structure of Vn . In each case of rank 1 to 7, we show Vn is rational and obtain a parametric representation of all rational points of Vn . For the case of rank ≥ 5, we define another variety which is birational to Vn , and parametrize rational points on Vn in these cases using this variety. The concrete proof is given only for the case of rank 6, because it is the most typical and richest and the other cases are treated more easily. In Section 4, for a given elliptic curve with generators whose rank ≤ 7, we specify the values of the parameters which are used in Section 3 to express the rational points of Vn . As an application, we reconstruct the example of rank 7 due to Grunewald and Zimmert [2] in Section 5. I would like to express my gratitude to Professor Fumio Hazama for his useful advice. And I also thank Professor Kenneth A. Ribet and Professor Robert Coleman for their stimulating conversation. 2. Generic case and its specialization. In this section, we construct an elliptic curve with rank n defined over the function field of an algebraic variety. Let E be an elliptic curve over a field k of characteristic 6= 2 defined by the following equation (1) E : y 2 = ax3 + bx2 + cx + d, and let f (x) be the right hand side of the equation of E. Then we can express E n by the simultaneous equation: yi2 = f (xi ) (i = 1, . . . , n). Let ιi the involution on E n defined by ι((xi , yi )) = (xi − yi ) (i = 1, . . . , n) and put Vn = E n /h(ι1 , . . . , ιn )i, then the function field of Vn is the set of the invariant elements of the function field of E n under the action of h(ι1 , . . . , ιn )i. Consequently, k(Vn ) = k(E n )h(ι1 ,... ,ιn )i = k(y1 y2 , . . . , y1 yn , x1 , . . . , xn ). Since (y1 yi+1 )2 = f (x1 )f (xi+1 ) holds for i = 1, . . . , n − 1, we find that Vn is defined by (2) yi2 = f (x1 )f (xi+1 ) (i = 1, . . . , n − 1). ELLIPTIC CURVES 191 (Here we rename y1 yi+1 as yi .) Let Eb be the twist of E by the quadratic extension k(E n )/k(Vn ). It is defined by the equation f (x1 )y 2 = f (x) (see [3, §4]). Let Eb (k(Vn )) be the group of k(Vn )-rational points on Eb . Theorem 2.1 (Hazama). If Endk (E) ∼ = Z, then the rank of Eb (k(Vn )) is n, and its generators are the following: yi (x1 , 1) xi+1 , (i = 1, . . . , n − 1). f (x1 ) Now, we can obtain a given elliptic curve with its generators by specializing the above twisted generic elliptic curve at a certain k-rational point on Vn as follows: Proposition 2.2. Let E be a given elliptic curve defined by the following equation E : y 2 = ax3 + bx2 + cx + d, and let (αi , βi ) (i = 1, . . . , n) be its independent generators. Let Eb be the twist of E by k(E n )/k(Vn ). Then E with these generators is obtained by specialization at the point (x1 , . . . , xn , y1 , . . . , yn−1 ) = (α1 , . . . , αn , β1 β2 , . . . , β1 βn ) on Vn . Proof. Put xi = αi (i = 1, . . . , n). Then Eb : f (x1 )y 2 = f (x) is isomorphic to y 2 = f (x) by the map (x, y) 7→ (x, β1y). Here the generators of twisted elliptic curve become (α1 , 1), αi+1 , ββi+1 (i = 1, . . . , n − 1). Therefore they 1 are mapped to (αi , βi ) (i = 1, . . . , n − 1). 3. The structure of the base space. In this section, we investigate the structure of the variety Vn defined in Section 1 in each case. The case of rank 1 is slightly different from the other cases, and can be treated easily. More precisely, the twisted elliptic curve (3) Eb : f (x1 )y 2 = ax3 + bx2 + cx + d, has a rational point (x1 , 1) ∈ Eb (k(x1 )), and it follows from Theorem 2.1 that it is of rank one as an elliptic curve over k(x1 ). But a generalization opposed to a specialization decreases the rank of an elliptic curve, hence Eb regarded as an elliptic curve over k(x1 , a, b, c, d) is of rank 1. As is seen from this argument for the case of rank 1, it is natural to regard Vn defined by (2) (n ≥ 2) as an algebraic variety defined over K = k(x1 , . . . , xn ). Therefore Vn is a 3-dimensional subvariety in the projective n+2 space Pn+2 with coordinates (a, b, c, d, y1 , . . . , yn−1 ), and Eb is regarded as a generic fiber of the elliptic fiber space defined by Equation (3) over Vn . 192 HIZURU YAMAGISHI Case of rank 2. Our V2 is defined by one quadratic equation with a rational point P1 (a, b, c, d, y1 ) = (0, 0, 0, 1, 1), hence V2 is K-rational and we can parametrize K-rational points on V2 as follows: Theorem 3.1. V2 is rational, in fact, each K-rational point on V2 is expressed as ((S + T )p1 , (S + T )p2 , (S + T )p3 , (S + T )p4 − ST, −ST ), where (p1 , p2 , p3 , p4 ) ∈ P3 and S = p1 x31 + p2 x21 + p3 x1 + p4 , T = p1 x32 + p2 x22 + p3 x2 + p4 . Case of rank 3. We recall two results which will be frequently used later. The first of them is classical and well-known (see [1], for example), but in view of the fundamental role played by it, we recall its proof. Lemma 3.2. Let V be a complete intersection of l quadrics in Pl−1 defined over k. Suppose that it contains a linear subvariety W k-isomorphic to Pl−1 . Then V is k-rational. Proof. There is a k-linear subvariety L k-isomorphic to Pn such that the intersection of W and L is empty. For any point P on L(k), let M be the variety spanned by {P } and W , which is k-isomorphic to Pl . Then we can express the intersection of the variety defined by i-th equation of V and M as the union of W and a k-linear subvariety Wi ∼ = Pl−1 . Since Wi (i = 1, . . . , l) are in M , we get a k-rational point Q as intersection of Wi (i = 1, . . . , l). The map ϕ : L → V defined by ϕ(P ) = Q gives a birational map from L(∼ = Pn−l ) to V . The next result is from an elementary linear algebra: Lemma 3.3. Let N be a given n × (n + 1) matrix and we denote the matrix removed i-th column by Ni . The homogeneous equation λ1 .. N . =0 λn λn+1 in Pn has a unique solution if N is of full rank. And then the solution is det(N1 ) λ1 .. .. . . i+1 λi = (−1) det(Ni ) . . .. .. . n+2 λn+1 (−1) det(Nn+1 ) ELLIPTIC CURVES 193 Now we show that V3 is birational to P3 . As V3 is a (2, 2)-intersection in P5 with coordinates (a, b, c, d, y1 , y2 ) and contains a line W defined by the equations x21 a + x1 b + c = 0, d = y1 = y2 = 0, therefore we can apply Lemma 3.2 and 3.3. Note that W is spanned by S1 = (1, 0, −x21 , 0, 0, 0), S2 = (0, 1, −x1 , 0, 0, 0). By a direct computation based on the map ϕ in the proof of Lemma 3.2, we obtain the following: Theorem 3.4. V3 is rational. Every K-rational point on V3 is given by (λ + νp1 , µ, −λx21 − µx1 , νp2 , νp3 , νp4 ), where (p1 , p2 , p3 , p4 ) ∈ P3 and (λ, µ, ν) = x31 x2 x3 (x3 − x2 )(x1 x3 − x2 x3 + x1 x2 )p21 − (x3 − x2 )(x41 − (x2 + x3 )x31 − x1 x2 x3 (x2 + x3 ) + x22 x23 )p1 p2 + (x3 − x2 )(x2 − x1 + x3 )p22 − x3 (x3 − x1 )p23 + x2 (x2 − x1 )p24 , − x51 x2 x3 (x23 − x22 )p21 − x21 (x1 + x2 )(x1 + x3 )(x3 − x2 )(x2 + x3 − x1 )p1 p2 − (x3 (x23 − x21 ) − x2 (x22 − x21 ))p22 + x3 (x23 − x21 )p33 − x2 (x22 − x21 )p24 , x2 x3 (p1 x31 + p2 )(x3 − x1 )(x2 − x1 )(x3 − x2 ) ∈ P2 . Case of rank 4. In this case, V4 contains the plane W defined by the equations x31 a + x21 b + x1 c + d = 0, y1 = y2 = y3 = 0. Hence by a similar argument to the one employed in the case of rank 3, we obtain the following: Theorem 3.5. V4 is rational. Every K-rational point on V4 is expressed as (λ + ρp1 , µ, ν, −(λx31 + µx21 + νx1 ), ρp2 , ρp3 , ρp4 ), where (p1 , p2 , p3 , p4 ) ∈ P3 and (λ, µ, ν, ρ) = − x31 p21 (x3 − x2 )(x4 − x2 )(x4 − x3 ) · (x21 (x2 + x3 + x4 ) − x1 (x2 x3 + x3 x4 + x2 x4 ) + x2 x3 x4 ) + p22 (x3 − x2 )(x4 − x1 )(x4 − x3 ) − (x2 − x1 )((x4 − x1 )(x4 − x2 )p23 − (x3 − x1 )(x3 − x2 )p24 , x61 p21 (x3 − x2 )(x4 − x2 )(x4 − x3 )(x2 + x3 + x4 ) − p22 (x3 − x1 )(x4 − x1 )(x4 − x3 )(x1 + x3 + x4 ) + p23 (x2 − x1 )(x4 − x1 )(x4 − x2 )(x1 + x2 + x4 ) 194 HIZURU YAMAGISHI − p24 (x2 − x1 )(x3 − x1 )(x3 − x2 )(x1 + x2 + x3 ), − x61 p21 (x3 − x2 )(x4 − x2 )(x4 − x3 )(x2 x3 + x3 x4 + x2 x4 ) + p22 (x3 − x1 )(x4 − x1 )(x4 − x3 )(x1 x3 + x3 x4 + x1 x4 ) − p23 (x2 − x1 )(x4 − x1 )(x4 − x2 )(x1 x2 + x2 x4 + x1 x4 ) + p24 (x2 − x1 )(x3 − x1 )(x3 − x2 )(x1 x2 + x2 x3 + x1 x3 ), p1 x31 (x2 − x1 )(x3 − x1 )(x3 − x2 )(x4 − x1 )(x4 − x2 )(x4 − x3 ) . Before proceeding to the case of rank ≥ 5, we state the following theorem. It can be proved by a similar argument to the one for [4, Theorem 2.1]. Theorem 3.6. Let Vn be the algebraic variety over K defined by (2) where we regard a, b, c, d and yi (i = 1, . . . , n − 1, n ≥ 5) as variables. Then Vn is K-birational to the variety V̄n defined by the equations 0 1 2 3 i 2 =0 (i = 4, . . . , n − 1) 2 2 2 2 Y Y1 Y2 Y3 Yi 0 0 1 2 3 i n−1 in P with coordinate (Y0 , . . . , Yn−1 ), where we write 2 2 2 2 2 Y0 Y1 Y2 Y3 Yi 1 1 1 1 1 α0 α1 α2 α3 αi for the determinant α02 α12 α22 α32 αi2 (i = 4, . . . , n − 1), and αi = α03 α13 α23 α33 α3 i 2 Y0 Y12 Y22 Y32 Yi2 xi+1 (i = 0, . . . , n − 1). Case of rank 5. In this case, V̄5 in Theorem 3.6 is defined by one quadric equation with a rational point P1 = (1, 1, 1, 1, 1). Hence we have the following theorem which is similar to Theorem 3.1: Theorem 3.7. V̄5 is rational. Every K-rational point on V̄5 is expressed as (2p1 S − T, 2p2 S − T, 2p3 S − T, 2p4 S − T, −T ), where (p1 , p2 , p3 , p4 ) ∈ P3 and 0 1 2 3 4 , S= p 1 p 2 p 3 p 4 0 0 1 2 3 4 2 2 2 2 . p p p p 0 1 2 3 4 Case of rank 6. In this case, V̄6 is a (2, 2)-intersection in P5 and contains a line W which is spanned by S1 = (1, 1, 1, 1, 1, 1), S2 = (α0 , . . . , α5 ). Therefore, we can apply Lemma 3.2. Let L be a linear subspace spanned by S3 S4 S5 S6 = = = = ( α4 − α5 , 0, 0, 0, ( 0, α4 − α5 , 0, 0, ( 0, 0, α4 − α5 , 0, ( 0, 0, 0, α4 − α5 , α5 − α0 , α5 − α1 , α5 − α2 , α5 − α3 , α0 − α4 ), α1 − α4 ), α2 − α4 ), α3 − α4 ). ELLIPTIC CURVES 195 For any point P = p1 S3 + p2 S4 + p3 S5 + p4 S6 on L, the point on M is represented by the form λS1 + µS2 + νP. We denote the i-th coordinate of P by P (i − 1). Then the equation of W1 ∩ W2 is easily seen to be λ N µ = 0, ν where N is a 2 × 3-matrix (Nij ) defined by 0 1 2 3 4 , N11 = 2 P (0) P (1) P (2) P (3) P (4) 0 1 2 3 4 , N12 = 2 α0 P (0) α1 P (1) α2 P (2) α3 P (3) α4 P (4) 0 1 2 3 4 , N13 = 2 2 2 2 P (0) P (1) P (2) P (3) P (4)2 0 1 2 3 5 N21 = 2 , P (0) P (1) P (2) P (3) P (5) 0 1 2 3 5 , N22 = 2 α0 P (0) α1 P (1) α2 P (2) α3 P (3) α5 P (5) 0 1 2 3 5 N23 = . P (0)2 P (1)2 P (2)2 P (3)2 P (5)2 Then by Lemma 3.3, λ det(N1 ) µ = − det(N2 ) , ν det(N3 ) where Ni is the 2 × 2-matrix with the i-th column removed from N . Hence the point on V̄6 which corresponds to P is λS1 + µS2 + νP , where (λ, µ, ν) = (det(N1 ), − det(N2 ), det(N3 )). Hence we have the following: Theorem 3.8. V̄6 is rational. Every K-rational point on V̄6 is expressed as λ + µα0 + νp1 (α4 − α5 ), λ + µα1 + νp2 (α4 − α5 ), λ + µα2 + νp3 (α4 − α5 ), λ + µα4 + ν 3 X pi+1 (α5 − αi ), λ + µα5 + ν i=0 where (p1 , p2 , p3 , p4 ) ∈ λ + µα3 + νp4 (α4 − α5 ), 3 X pi+1 (αi − α4 ) i=0 P3 , (λ, µ, ν) = (N12 N23 − N13 N22 , −N11 N23 + N13 N21 , N11 N22 − N12 N21 ) with Nij as above. 196 HIZURU YAMAGISHI Case of rank 7. In this case, V̄7 is a (2, 2, 2)-intersection in P6 . Unfortunately, this is not rational and only unirational [1]. To remedy this situation, we consider E : y 2 = ax4 + bx3 + cx2 + dx + e instead of (1). Let f (x) be the right hand side of the equation of E and V7 the variety defined by the equation yi2 = f (x1 )f (xi+1 ) (i = 1, . . . , 6). Repeating a similar argument to the one given above, we obtain the variety which is birational to V7 defined by the equation, 0 1 2 3 4 i 2 =0 (i = 5, 6, 7). 2 2 2 2 2 Y Y1 Y2 Y3 Y4 Yi 0 We call this V̄7 . This is in P7 and its dimension is 4. Note that V̄7 contains a plane W spanned by S1 = (1, . . . , 1), S2 = (α0 , . . . , α7 ), S3 = (α02 , . . . , α72 ). Hence we are in the same situation as the case of rank 6. Thus we have the following theorem which can be proved similarly: Theorem 3.9. V̄7 is rational. Every K-rational point on V̄7 is expressed as λ + µα0 + να02 , λ + µα1 + να12 , λ + µα2 + να22 , λ + µα3 + να32 + ρp1 , λ + µα4 + να42 + ρp2 , λ + µα5 + να52 + ρp3 , λ + µα6 + να62 + ρp4 , λ + µα7 + να72 + ρp5 , where (p1 , p2 , p3 , p4 , p5 ) ∈ P4 and (λ, µ, ν) = (det(N1 ), − det(N2 ), det(N3 ), − det(N4 )) with Ni (i = 1, . . . , 4) as following: 0 1 2 N11 = 2 0 0 0 0 1 2 N12 = 2 0 0 0 0 1 2 N13 = 2 0 0 0 0 1 2 N14 = 0 0 0 0 1 2 N21 = 2 0 0 0 0 1 2 N22 = 2 0 0 0 0 1 2 N23 = 2 0 0 0 3 4 5 , p 1 p2 p 3 3 4 5 , α3 p1 α4 p2 α5 p3 3 4 5 , α32 p1 α42 p2 α52 p3 3 4 5 , p21 p22 p23 3 4 6 , p1 p2 p4 3 4 6 , α3 p1 α4 p2 α6 p4 3 4 6 , α32 p1 α42 p2 α62 p4 ELLIPTIC CURVES N24 N31 N32 N33 N34 0 = 0 0 = 2 0 0 = 2 0 0 = 2 0 0 = 0 197 1 2 3 4 6 , 0 0 p21 p22 p24 1 2 3 4 7 , 0 0 p 1 p 2 p5 1 2 3 4 7 , 0 0 α3 p1 α4 p2 α7 p5 1 2 3 4 7 , 0 0 α32 p1 α42 p2 α72 p5 1 2 3 4 7 . 0 0 p21 p22 p25 Remark 3.10. In the case of n ≤ 4, one can write down immediately the defining equation of elliptic curve which corresponds to a point on Vn . In the case of n ≥ 5, let P = (ȳ0 , . . . , ȳn−1 ) be a rational point on V̄n . Then, the defining equation of the elliptic curve which corresponds to P is 1 1 1 1 1 α0 α1 α2 α3 x 2 α α2 α2 α2 x2 = 0. 1 2 3 03 α0 α13 α23 α33 x3 2 ȳ0 ȳ12 ȳ22 ȳ32 y 2 This is obtained by tracing the birational map between Vn and V̄n . 4. The value of the parameter which corresponds to a given elliptic curve. In view of Proposition 2.2, every elliptic curve with rank n should correspond to a point of Vn . In this section, we give the values of the parameters for this point of Vn (2 ≤ n ≤ 7). let E be an elliptic curve defined by the following equation y 2 = ax3 + bx2 + cx + d with independent generators (xi , yi ) (i = 1, . . . , r, 2 ≤ r ≤ 7). Then we have the values of parameters as follows: 198 HIZURU YAMAGISHI r=2 (p1 , p2 , p3 , p4 ) = (a, b, c, d − y1 y2 ), r=3 (p1 , p2 , p3 , p4 ) = ax21 + bx1 + c , d, y1 y2 , y1 y3 , x21 r=4 (p1 , p2 , p3 , p4 ) = y1 , y2 , y3 , y4 , x31 r=5 (p1 , p2 , p3 , p4 ) = (y1 − y5 , y2 − y5 , y3 − y5 , y4 − y5 ), r=6 pj 6 X =− 6 X ! (x6 − xi+1 ) i=1 + − 6 X i=1 6 X + (x5 − xi+1 ) ! (x5 − xi+1 ) (x6 − xi+1 ) xi+1 (x6 − xi+1 ) ! xi+1 (x6 − xi+1 ) 6 X ! (x6 − xi+1 ) − 6 X ! (x6 − xi+1 ) i=1 − ! yi+1 (x5 − xi+1 ) 6 X i=1 6 X ! yi+1 (x6 − xi+1 ) ! yi+1 (x6 − xi+1 ) i=1 ! xi+1 (x5 − xi+1 ) i=1 i=1 + yj ! i=1 6 X ! 6 X i=1 i=1 6 X i=1 + xj xi+1 (x6 − xi+1 ) i=1 6 X ! i=1 6 X ! 6 X i=1 6 X ! (x6 − xi+1 ) ! (x5 − xi+1 ) i=1 6 X i=1 6 X ! (x5 − xi+1 ) ! (x6 − xi+1 ) i=1 6 X i=1 6 X i=1 6 X ! yi+1 (x6 − xi+1 ) ! yi+1 (x5 − xi+1 ) !! yi+1 (x6 i=1 6 X − xi+1 ) ! xi+1 (x6 − xi+1 ) i=1 6 X !! xi+1 (x5 − xi+1 ) , i=1 (j = 1, . . . , 4). In the case r = 7, let E be an elliptic curve defined by the following equation y 2 = ax4 + bx3 + cx2 + dx + e ELLIPTIC CURVES 199 with independent generators (xi , yi ) (i = 1, . . . , 8). Then the values of the parameters are given by the formula pi = (x3 − x2 )(y1 (x2 x3 − x3 xi+3 − x2 xi+3 + x2i+3 ) − yi+3 x2 x3 ) − (x3 − x1 )(y2 (x1 x3 − x3 xi+3 − x1 xi+3 + x2i+3 ) − yi+3 x1 x3 ) + (x2 − x1 )(y3 (x1 x2 − x2 xi+3 − x2 xi+3 + x2i+3 ) − yi+3 x1 x2 ), (i = 1, . . . , 5). 5. Examples. In the previous sections, we have given a parametric representation of Vn (1 ≤ n ≤ 7). Therefore we will get as many elliptic curves with specified rank as we like by specialization at rational points on Vn . As an example, we give the values of parameters which enable one to obtain the elliptic curve with rank 7 in [2, Corollary C]. The elliptic curve is y 2 = x3 − 1717730532x + 27401746395780 with generators (24144, 56466), (23562, 97182), (23736, 50022), (24840, 245430), (25422, 404082), (23793, 34119), (26121, 596187). The values of the parameters are (p1 , p2 , p3 , p4 , p5 ) = (47822467248393469632, 66206014691795224675, 104521834162171114920, 76522282208132178600, 101559585548776675320), (x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 ) 1 1 1 1 1 1 1 , − , − , , , − , , 0 . = 234 348 174 930 1512 117 2211 References [1] P.A. Beauville, Variétés de Prym et Jacobiennes Intermédiaires, Ann. Scient Éc. Norm. Sup., 4 (1977), 309-391. [2] F.J. Grunewald and R. Zimmert, Über einige rationale elliptische Kurven mit freien Rang ≥ 8, J. Reine. Angew. Math., 296 (1977), 100-107. [3] F. Hazama, Rational points on certain abelian varieties over function fields, J. Number Theory, 50 (1995), 278-285. [4] H. Yamagishi, On abelian surfaces with rank 12 associated to certain K3 surfaces, in ‘Algebraic cycles and related topics’, Ed. F. Hazama, World Scientific, (1995), 93-102. [5] , Geometric construction of elliptic curves with given Mordell-Weil rank and j-invariant, (submitted to Proceedings of the American Mathematical Society). 200 [6] HIZURU YAMAGISHI , A construction of families of elliptic curves of rank 6 with a nontrivial twotorsion point, (submitted to Journal of Number Theory). Received February 5, 1998 and revised June 24, 1998. Tokyo Denki University Saitama, 350-0394 Japan E-mail address: [email protected] Guidelines for Authors Authors may submit manuscripts at pjm.math.berkeley.edu/about/journal/submissions.html and choose an editor at that time. Exceptionally, a paper may be submitted in hard copy to one of the editors; authors should keep a copy. By submitting a manuscript you assert that it is original and is not under consideration for publication elsewhere. Instructions on manuscript preparation are provided below. 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Failure to acknowledge the receipt of proofs or to return corrections within the requested deadline may cause publication to be postponed. Volume 191 No. 1 November 1999 Explicit Cayley triples in real forms of E 7 D RAGOMIR Ž. Ð OKOVI Ć 1 25 Ordering the braid groups R. F ENN , M.T. G REENE , D. ROLFSEN , C. ROURKE AND B. W IEST 49 Numerical semigroups generated by intervals P.A. G ARCÍA -S ÁNCHEZ AND J.C. ROSALES 75 A Selberg integral formula and applications L OUKAS G RAFAKOS AND C ARLO M ORPURGO 85 Ascent and descent for finite sequences of commuting endomorphisms L UZIUS G RUNENFELDER 95 123 Cohomology of complete intersections in toric varieties A NVAR R. M AVLYUTOV 133 Connected sums of self-dual manifolds and equivariant relative smoothings H ENRIK P EDERSEN AND YAT S UN P OON 145 Some summations of q-series by telescoping M.V. S UBBARAO AND A. 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