Volume 191
No. 1
November 1999
Pacific Journal of Mathematics
PACIFIC JOURNAL OF MATHEMATICS
Pacific
Journal of
Mathematics
1999
Vol. 191, No. 1
Volume 191
No. 1
November 1999
PACIFIC JOURNAL OF MATHEMATICS
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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
ñ[ 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
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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.
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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]
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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. V ERMA
173
Dehornoy’s ordering of the braid groups extends the subword ordering
B ERT W IEST
183
A unified method of construction of elliptic curves with high Mordell–Weil
rank
H IZURU YAMAGISHI
Vol. 191, No. 1
Topology versus Chern numbers for complex 3-folds
C LAUDE L E B RUN
Pacific
Journal of
Mathematics
1999
On the index formula for singular surfaces
B. F EDOSOV , B.-W. S CHULZE AND N. TARKHANOV
Pacific Journal of Mathematics
PACIFIC JOURNAL OF MATHEMATICS
189
Volume 191
No. 1
November 1999