ESI
The Erwin Schrodinger International
Institute for Mathematical Physics
Boltzmanngasse 9
A-1090 Wien, Austria
Compatible Almost Complex Structures on
Quaternion{Kahler Manifolds
D.V. Alekseevsky
S. Marchiafava
M. Pontecorvo
Vienna, Preprint ESI 419 (1997)
Supported by Federal Ministry of Science and Research, Austria
Available via http://www.esi.ac.at
January 10, 1997
COMPATIBLE ALMOST COMPLEX STRUCTURES
ON QUATERNION-KA HLER MANIFOLDS
D.V. ALEKSEEVSKY1 - S. MARCHIAFAVA2 - M. PONTECORVO3
Sophus Lie Center (Moscow)1 - Universita' di Roma I2 - Universita di Roma Tre3
Abstract. Let (M 4n ; g; Q) be a quaternion-Kahler manifold with reduced scalar curvature = K=4n(n +2). Suppose J is an almost complex structure which is compatible
with the quaternionic structure Q and let = ? J be the Lee form of J . We prove
the following local results: 1) if J is conformally symplectic then it is parallel and
= 0; 2) if J is cosymplectic then 0 and = 0 if and only if J is parallel; 3) if J
is integrable then d is Q-Hermitian and harmonic; 4) if J is conformally balanced but
not symplectic then there exists an associated non-zero Killing vector eld. We prove
also that any closed self-dual 2-form ! 2 2+ = g Q 2 is parallel.
When M 4n is compact our main global results are the following: 1) if > 0 and
(M 4n ; g) is dierent from the Grassmannian G2 (C m ) of 2-planes of C m then there
exists no compatible almost complex structure J ; 2) if the rst Chern class c1 (J ) c1 (T M ) = 0 for the almost complex structure J on the tangent bundle T M then
= 0; 3) if 6= 0 then there exists no compatible complex structure J ; 4) if = 0 a
compatible complex structure J is parallel. The last two results have been proved in
[P2] by twistor methods.
In x5 and x6 we compute the Laplacian and the (real and quaternionic) Weyl tensors
acting on self-dual 2-forms, by relating to the work of Gauduchon in [G1].
1. Introduction and main results
In a previous paper ([AMP]) we studied several problems concerning almost complex structures which are dened on an almost quaternionic manifold (M 4n ; Q) and
which are compatible with the almost quaternionic structure Q. We recall that an
almost quaternionic structure Q on M 4n is a rank-3 subbundle Q End(TM ) which
is locally spanned by almost hypercomplex structures H = (J ); such a locally dened
triple H = (J), where J2 = ?id and J1J2 = ?J2J1 = J3, is called an admissible
4n
basis of Q. An almost
P complex structure J on M is compatible with Q if it can be
written as J = cJ with respect to any admissible basis H = (J) for suitable
1991 Mathematics Subject Classication. 53C10 - 32C10.
Work done under the program of G.N.S.A.G.A. of C.N.R. and partially supported by M.U.R.S.T
(Italy) and E.S.I. (Vienna).
Typeset by AMS-TEX
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ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
functions c. By referring to the twistor bration t : Z ?! M 4n one can say equivalently that J is a section of the twistor space Z , J : M ?! Z , see [S3, pag. 130] and
[AMP].
In this work we will consider the stronger notion of a quaternion-Kahler manifold
(M 4n ; g; Q) . This means that g is a Riemannian metric on M which is Q-Hermitian
(i.e. any endomorphism of Q is g-skew-symmetric) and, moreover, that the LeviCivita connection of g preserves Q. This is equivalent to the statement that the
holonomy group of g is contained in Sp(n) Sp(1). Notice that every oriented Riemannian 4-manifold satises the above conditions. For this reason we will assume that
the quaternionic dimension n 2 and we will systematically treat the 4-dimensional
case in separate statements.
For the basic properties of quaternion-Kahler manifolds we will refer to [S1] and
[S2]. Let us just recall here that any quaternion-Kahler manifold (M 4n ; g; Q) with
n 2 is automatically Einstein. When the scalar curvature K of g is positive
a complete quaternion-Kahler manifold is always compact and simply connected;
the only known examples are the symmetric spaces of Wolf [W]. When K = 0 the
bundle Q is locally parallelizable so that the universal covering of M is a hyperKahler manifold.
We will assume that (M 4n ; g; Q) , n 2, is a quaternion-Kahler manifold and
discuss various properties of a compatible almost complex structure J , under several
assumptions (see denitions 2.2) both from local and global point of view.
Our main results about existence of local compatible almost complex structures
can be summarized as follows.
Let (M 4n ; g; Q) be a quaternion-Kahler manifold with reduced scalar curvature and J an almost complex structure which is compatible with the quaternionic structure
Q on M 4n . Let F = g J , = ?F J be the Kahler form and , respectively, the
Lee form of J . Then:
1) if J is locally conformally symplectic (that is dF = ^ F for some 1-form )
then it is parallel and = 0.
2) if J is cosymplectic (that is = 0) then 0 and = 0 if and only if J is
parallel.
3) if J is integrable then the dierential d of the Lee form is Q-Hermitian and
harmonic.
4) if J is conformally balanced (i.e. J is integrable and is exact) but not symplectic then there exists a non-zero Killing vector eld associated to J .
The proofs of the statements will be given through Theorems 2.4, 3.1, 3.2 and
Propositions 4.6, 4.11.
A further local result concerns self-dual forms on M 4n , that is sections of the 3dimensional subbundle 2+ = g Q of 2 which corresponds to Q by means of the
metric g. In Theorem 3.1 we prove that any closed self-dual 2-form ! is parallel .
Our main global results can be stated as follows.
ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS
3
Let (M 4n ; g; Q) be compact, then
1) if > 0 and (M 4n ; g) is dierent from the Grassmannian G2(C m ) of 2-planes
in C m , then there exists no compatible almost complex structure J .
2) if the rst Chern class c1(J ) c1(TJ(1;0)M ) = 0 for some compatible almost
complex structure J , then = 0.
3) if 6= 0 there exists no compatible complex structure J .
4) if = 0 a compatible complex structure J is parallel.
The last two results have been proved in [P2] by twistor methods.
The proof of these statements will be given throughout Proposition 3.7 and its
Corollary 3.8, Theorem 4.3, Proposition 4.6 and Theorem 3.10 where we establish
the relationship between the rst Chern classes c1(J ) of (TM; J ) and the rst Chern
class c1(J ?) of the complex line bundle J ? orthogonal to J in Q.
In paragraphs 5 and 6 we compute the Laplacian and the (real and quaternionic)
Weyl tensors acting on self-dual 2-forms. This part is very much related to work of
Gauduchon [G1].
2. Preliminary results
Notations and denitions. Let (M n ; g; Q) be a quaternion-Kahler manifold,
with n 2. We recall that the Riemannian metric g is Hermitian with respect
4
to Q, that is
g(JX; Y ) + g(X; JY ) = 0
for all X; Y 2 TM 4n and any J 2 Q, and also it is Einstein, that is
Ric(g) = 4Kn g
where K is the scalar curvature.
We dene the bundle 2+ ?! M of self-dual 2-forms as the 3-dimensional bundle
associated to Q by the metric g, that is 2+jx = (g Q)x ; x 2 M . If H = (J) is an
admissible basis of Q the Kahler forms F of the almost Hermitian structures (J; g)
F = g J = g(J; )
( = 1; 2; 3)
represent a (local) frame for 2+. The notation is consistent with the one used for an
oriented 4-dimensional manifold (see also [AMP, section 5]).
The Levi-Civita connection r rg of g is a quaternionic connection, that is for
any admissible basis H = (J):
rJ = ! J ? ! J
(2:1:1)
where the ! , = 1; 2; 3 are 1-forms and (; ; ) is a cyclic permutation of (1,2,3).
Hence
dF = ! ^ F ? ! ^ F
(2:1:2)
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ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
since rF = g(rJ; ).
We recall that the following fundamental identities hold:
d! + ! ^ ! = ?F
( = 1; 2; 3)
(2:2)
where = K=4n(n + 2) is the reduced scalar curvature of g. Moreover
F = 21n Tr[J R(X; Y )]
( = 1; 2; 3)
(2:3)
where R is the curvature tensor of r (see [AM2]).
We recall also that on M 4n there is a globally dened fundamental 4-form ,
=
X
F ^ F
which is parallel and non degenerate, see [Be, pag.419].
Remark 2.1. For n = 1, an oriented Riemannian 4-manifold (M 4 ; g) always satises (2.1.1), (2.1.2). The fundamental identities (2.2) hold if g is anti-self-dual and
Einstein. M
Now we dene dierent types of compatible almost complex structures J 2 ?(Q)
on a quaternion-Kahler manifold (M 4n ; g; Q) .
Denitions 2.2. Let J be an almost complex structure on the Riemannian manifold
(M 4n ; g). Then J is said to be:
(1) parallel if rJ = 0 where r is the Levi-Civita connection of g.
Moreover let assume that g is J -Hermitian. Then J is called to be
(2) symplectic if the Kahler form F = g J is closed, dF = 0;
(3) locally conformally symplectic if dF = ^ F for some 1-form ;
(4) cosymplectic if F is coclosed, F = 0 .
Remark 2.3. If n = 1 dF = ^ F always holds (see also (2.5) below); for n 2 the
same equality does not hold in general and it implies that is closed. M
Parallel complex structures on a quaternion-Kahler manifold. The following
theorem describes a parallel complex structure on a quaternion-Kahler manifold.
Theorem 2.4. Let (M 4n ; g; Q) be a simply connected complete irreducible quaternion-Kahler manifold with reduced scalar curvature . Let J be a parallel complex
structure on M 4n.
P
P
(1) If J is compatible then = 0, and J = 3=1 a J where a = const; a2 =
1 and H = (J) is a parallel hypercomplex structure,
(2) otherwise (M 4n ; g) is isometric to the quaternionic symmetric space G2 (C m )
or to its dual non compact symmetric space, and J is the unique parallel
complex structure.
ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS
5
Proof. Remark that the parallel complex structures on M 4n correspond to the complex structures in the tangent space of a point x which are invariant under the holonomy group Hx. If = 0 the manifold (M 4n ; g; Q) has the holonomy group Sp(n) and
the rst statement holds. If 6= 0 then either the holonomy group Hx = Sp(n) Sp(1)
or the manifold is locally symmetric (see [Be]): the rst case cannot occur since there
is no Sp(n) Sp(1) -invariant complex structure , and investigation of all holonomy
groups of symmetric quaternion-Kahler manifolds gives the second statement. Some local formulas. Now we derive some formulas which will be useful for studying cosymplectic and integrable almost complex structures.
For a tensor T = (Tijk ), where Tijk are the components of T with respect to
an orthonormal basis, we denote
kT k2 =
X
i;j; ;k
(Tijk )2
(In the terminology of Gauduchon k k is the tensorial norm j jT ; [G1, p. 5]).
For example,
kFk2 = 4n
( = 1; 2; 3)
From now on we will assume that J is a compatible almost complex structure on
(M 4n ; g; Q) with Kahler form F , H = (J = J1; J2; J3) is a local admissible basis and
! are the corresponding 1-forms. We dene the Lee form of J by
= ?(F ) J
(2:4)
where is the codierential. It is known that
d(F 2n?1) = ^ F 2n?1
(2:5)
Proposition 2.5. For a compatible almost complex structure J on a quaternionKahler manifold the following formulas hold:
krF k2 = 4n(k!2k2 + k!3k2 )
(2:6)
kdF k2 = 12(n ? 1)(k!2 k2 + k!3 k2) + 6k!2 J2 + !3 J3k2
(2:7)
1 kdF k2 ? 1 krF k2 = ?k! J ? ! J k2
2
2
3
3
6
2
= ! 2 J 2 + ! 3 J3
Moreover, by denoting h; i = g(; ) to simplify,
= 4n ? 2h!2 J2; !3 J3i
(2:8)
(2:9)
(2:10)
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ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
and
1 kdF k2 ? 1 krF k2 + kk2 + 2 = 8n
6
2
(2:11)
Remarks 2.6. 1) The inequality kdF k2 3krF k2 which comes from (2.8) was previously established in [G1]. The equality holds if and only if J is integrable (see also
Lemma 4.1 below).
2) For n = 1, the (2.7) still holds: kdF k2 = 6k!2 J2 + !3 J3k2.
3) For n = 1 the (2.9) still holds since dF1 = ^ F1 and, as it is easy to see, the
identity ? ^ F1 + !3 ^ F2 ? !2 ^ F3 = 0 holds if and only if ? J1 + !3 J2 ? !2 J3 = 0.
The (2.10), (2.11) hold under the hypothesis that the metric g is anti-self-dual and
Einstein. M
Proof. Since rF = g(rJ1; ) the (2.1.1) gives rF = !3 J2 ? !2 J3. We compute
krF k2, by using an orthonormal frame (E1 ; ; E4n). By denoting h; i = g(; ),
we have
X
krF k2 = [(rEi F )(Ej ; Ek )]2
=
=
i;j;k
X
i;j;k
X
i;j;k
[!3(Ei )hJ2Ej ; Ek i ? !2(Ei )hJ3 Ej ; Ek i]2
[!32(Ei)hJ2 Ej ; Ek i2 + !22(Ei )hJ3Ej ; Ek i2
? 2!2(Ei )!3(Ei )hJ2 Ej ; Ek ihJ3 Ej ; Ek i]
= 4n(k!2k2 + k!3k2 )
P hJ E ; E i2 = kJ E k2 = kE k2 and
For
last
identity
we
used
j
j
P hJ E ; E ihJ E ; E i = hkJ E2 ;jJ Ek i = 0. 2Hence
the
(2.6)
is proved.
2
j
k
3
j
k
2
j
3
j
k
Now we prove (2.7). We have
kdF k2 =
=
=
X
i;j;k
X
i;j;k
X
i;j;k
dF (Ei; Ej ; Ek )2
[(rEi F )(Ej ; Ek ) + (rEk F )(Ei ; Ej ) + (rEj F )(Ek ; Ei)]2
[!3(Ei)hJ2 Ej ; Ek i + !3(Ek )hJ2Ei ; Ej i + !3(Ej )hJ2 Ek ; Ei i
? !2(Ei )hJ3Ej ; Ek i ? !2(Ek )hJ3Ei ; Ej i ? !2(Ej )hJ3 Ek ; Ej i]2
By computing the square of the expression in brackets, after a long but straightforward calculation, we get
kdF k2 = (12n ? 6)(k!2 k2 + k!3k2) + 12h!2 J2; !3 J3i
ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS
7
This implies (2.7). The (2.8) follows directly from (2.6), (2.7). For a vector eld X
and an orthonormal frame Ei; i = 1; : : : ; 4n, we have
(F )(JX ) = ?
=?
X
X
i
h(rEi J1)Ei ; J1X i = ?
!3 (Ei)hJ2 Ei ; J1X i +
i
that is
X
i
X
i
h!3(Ei )J2 Ei ? !2(Ei )J3Ei ; J1X i
!2 (Ei)hJ3 Ei ; J1X i = ?!3(J3X ) ? !2 (J2X )
F = !3 J2 ? !2 J3
Hence (2.9) follows.
Now we compute . We have
rX = (rX !2) J2 + (rX !3 ) J3
+ !2 (!1 (X )J3 ? !3 (X )J1 ) + !3 (!2 (X )J1 ? !1(X )J2 )
Hence
= ?
X
i
[(rEi !2 )(J2Ei ) + (rEi !3 )(J3Ei )
? !1(Ei )(!2 J2 + !3 J3)(J1 Ei) + 2!2 (Ei)!3 (J1Ei )]
P
(2:12)
P
since i ?!3(Ei )!2(J1 Ei) = i !3 (J1Ei)!2 (Ei ). Now let take into account that
(rX !2 )Y ? (rY !2 )X = d!2(X; Y )
and hence, putting X = Ei and Y = J2Ei and summing up over i,
X
i
that is
[(rEi!2)(J2 Ei ) ? (rJ2Ei !2)(Ei )] =
2
Analogously
2
X
X
i
i
(rEi !2 )(J2Ei ) =
X
X
i
i
(rEi !3 )(J3Ei ) =
X
i
d!2(Ei ; J2Ei)
d!2(Ei ; J2Ei)
d!3(Ei ; J3Ei)
By the second of integrability conditions (2.2) one has
X
i
that is
X
i
d!2 (Ei; J2 Ei) =
d!2(Ei ; J2Ei ) =
X
i
X
i
[?(!3 ^ !1 )(Ei ; J2Ei) + hEi ; J22Eii]
[?!3(Ei )!1 (J2Ei) + !1(Ei )!3(J2 Ei)] ? 4n
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ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
Analogously, one has
X
i
d!3(Ei ; J3Ei ) =
Hence
2 = ?
X
i
X
i
[?!1(Ei )!2 (J3Ei) + !2(Ei )!1(J3 Ei)] ? 4n
[?!3(Ei )!1(J2 Ei) + !1 (Ei )!3(J2 Ei)
? !1(Ei )!2 (J3Ei ) + !2(Ei )!1 (J3Ei)
? 8n ? 2!1(Ei )(!2 J2 + !3 J3)(J1 Ei) + 4!2 (Ei)!3 (J1Ei )]
and (2.10) follows immediately. Since by (2.8), (2.9) one has 61 kdF k2 ? 21 krF k2 =
?k!2 J2 ? !3 J3k2 and kk2 = k!2 J2 + !3 J3k2 it follows
1 kdF k2 ? 1 krF k2 + kk2 = 4h! J ; ! J i
2
2
3
3
6
2
and, by (2.10), we get (2.11). 3. Compatible almost complex and almost hypercomplex structures
(Conformally) symplectic structures.
Theorem 3.1. Let (M n ; g; Q) be a quaternion-Kahler manifold with n 2. Then,
4
(1) every compatible locally conformally symplectic almost complex structure J
is parallel.
(2) every closed self-dual 2-form ! is parallel
In both cases = 0.
Proof. If J is a compatible locally conformally symplectic almost complex structure then dF = ^ F . Choose an admissible basis H = (J = J1; J2; J3) of Q
and denote by F, = 1; 2; 3, the corresponding Kahler forms. Then by (2.1.2)
dF = !3 ^ F2 ? !2 ^ F3 , hence ^ F ? !3 ^ F2 + !2 ^ F3 = 0. By [ABM, Lemma 1 p.125]
it follows that = !2 = !3 = 0 so that J is parallel and = 0. This proves the rst
statement. To prove the second statement we put M = fp 2 M j! 6= 0g. On M we
can write ! = fF where f is a never zero smooth function and F is the Kahler form
of a compatible almost complex structure . Without loss of generality, we may
assume that f > 0. Then we have 0 = d! = df ^ F + fdF and dF = ?d log f ^ F .
Now the rst statement implies that f is a constant and F and ! are parallel on M
and, hence, on M . ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS
9
Cosymplectic almost complex structures.
Theorem 3.2. Let J be a compatible cosymplectic almost complex structure on the
quaternion-Kahler manifold (M n ; g; Q) , that is = 0. Then 0 and k! k =
k! k = const = ?2n . Moreover = 0 if and only if J is parallel.
Proof. From (2.9) = 0 implies ! J = ?! J and in particular k! k = k! k.
4
3
2
2
Then, by (2.10) we have
3
3
2
2
2
2
3
0 = 4n + 2k!2k2
and conclusion is immediate By combining the expression (2.9) of given in Proposition 2.5 with a result of
[AMP, proposition 4.1] it is easy to deduce the following Proposition which for n = 1
was proved by S. Salamon in [S2, Prop. 1].
Proposition 3.3. Let (M 4n ; g; Q) be a quaternion-Kahler manifold and H = (J )
an admissible basis of Q:
(1) If J1; J2 are cosymplectic then J3 is integrable;
(2) If J1 is cosymplectic and J2 is integrable then J3 is cosymplectic;
(3) If J1; J2 are integrable then J3 is integrable.
Proof. We prove (1) as follows. Assume J1; J2 cosymplectic: then by (2.9) !2 J2 +
!3 J3 = !3 J3 + !1 J1 = 0 and hence !1 J1 = !2 J2 which is equivalent to the
integrability of J3 by proposition 4.1 of [AMP] (see also Lemma 4.1 below). To prove
(2) we remark that by hypothesis one has !2 J2 + !3 J3 = 0 and !1 J1 = !3 J3.
It implies !1 J1 + !2 J2 = 0, that is J3 is cosymplectic. (3) was proved in [AMP,
Remark 2.6]. Now we associate to an almost complex structure J compatible with Q a globally
dened closed 2-form J by the formula
J = d!1
(3.1)
for any admissible basis H = (J1 = J; J2; J3). If H 0 = (J10 = J; J20 ; J30 ) is another
such admissible basis and (!10 ; !20 ; !30 ) are corresponding 1-forms, then
J20 = cos'J2 + sen'J3 ; J30 = ?sen'J2 + cos'J3
which implies
!10 = !1 + d':
This shows that J is globally dened and depends only on J .
Proposition 3.4. Let J be a compatible almost complex structure. Assume that
= 0, that is J is cosymplectic. If < 0, then the 2-form J is a symplectic
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ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
form which tames J , that is g0 := J (; J ) is a Riemannian metric (which is quasiKahlerian with respect to J ).
Proof. Since by hypothesis !3 = !2 J1, for = 1 the fundamental identity (2.2)
gives
d!1 = ?!2 ^ (!2 J ) ? F
hence
d!1(; J ) = !2 !2 + (!2 J ) (!2 J ) ? g
and the conclusion follows immediately. Compatible almost complex and almost hypercomplex structures on a
compact quaternion-Kahler manifold. We will now relate the 2-form J to
the complex line bundle J ? orthogonal to J in Q.
Proposition 3.5. Let (M 4n ; g; Q) be a quaternion-Kahler manifold and J a compatible almost complex structure. Then the linear connection r0 on the bundle Q which
is dened by
r0X = rX ? 21 J rX J rX ? 12 [!2(X )J2 + !3(X )J3 ]
is Riemannian and preserves J (r0 is the rst canonical connection of the almost
Hermitian structure (g; J ), [G3, pag. 31]). The 2-form J is the curvature form of
the Riemannian connection induced by r0 on the complex line bundle J ? orthogonal
to J in Q.
When M 4n is compact the 2-form 21 J represents the Chern class c1 of J ? and
the following conditions are equivalent:
(1) The cohomology class c1(J ?) = [ 21 J ] 2 H 2 (M; Z) vanishes.
(2) The rst Chern class c1(J ) c1(TM ) of the tangent bundle TM with respect
to the almost complex structure J vanishes.
(3) There exists a global admissible basis H = (J1 = J; J2; J3) of Q on M 4n.
Proof. Let X; Y be any two vector elds on M . Since (rJ )J + J (rJ ) = 0, it is easy
to see that
g((rX J )Y; JZ ) + g(JY; (rX J )Z ) = 0
Hence r0 is Riemannian. Moreover,
(r0X J )(Y ) = (rX J )(Y ) ? 21 (J rX J )(JY ) + 21 J 2(rX J )(Y )
= (rX J )(Y ) + 21 (J 2rX J )(Y ) + 21 (J 2 rX J )(Y ) = 0
that is r0 leaves J invariant. Hence we can think r0 as a connection for the 2-plane
bundle J ? orthogonal complement to the line-bundle RJ in Q. For any admissible
ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS
11
basis H = (J1 = J; J2; J3) we have a local orthonormal frame (J2; J3) for J ? and the
following identities hold
r0X J2 = !1 (X )J3
;
r0X J3 = ?!1(X )J2
(3.2)
This shows that the connection r0 preserves the complex structure I on J ? , induced
by J :
I (J2) = J3 ; I (J3 ) = ?J2:
Then the matrix 1-form of the connection r0 with respect to the local frame (J2 ; J3)
and matrix 2-form of the curvature are given by
! = !01 ?0!1
1
; = d!1 ?d!
0 :
0
Hence, the 2-form J = 21 d!1 is the Chern form of r0, as to be proved. To prove
that conditions (1) and (3) are equivalent we observe that, from the classication of
S 1-bundles over M 4n by elements of H 2 (M; Z), one has c1(J ?) = 0 if and only if J ?
is trivial or, equivalently, if and only if there exists an admissible basis H = (J1 =
J; J2; J3 ) of Q = RJ J ? globally dened on M 4n . The equivalence of (1) and (2)
will follows immediately from (3.6) of Theorem 3.10 below. Before giving an application of the last result we recall the following result which
was proved in [AM2].
Theorem 3.6 ([AM2]). Let (M 4n ; g; Q) , n > 1, be a quaternion-Kahler manifold
with the reduced scalar curvature . Assume that there exists a (globally dened)
almost hypercomplex structure H = (J) which generates Q.
Then either
(1) = 0, that is (M 4n ; g) is a locally hyper-Kahler manifold.
or
P
(2) the fundamental 4-form = F ^ F is exact.
For a compact M 4n only (1) is possible.
Proof. By identities (2.2) one has
2 =
X
= d(
(d! + ! ^ ! ) ^ (d! + ! ^ ! )
X
! ^ d! + 2!1 ^ !2 ^ !3)
wherePthe 1-forms ! are globally dened. Hence, if 6= 0 the fundamental 4-form
= F ^ F is exact. The last case cannot occur if M 4n is compact, since the
form is parallel and, hence, harmonic. 12
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
Proposition 3.7. Let (M n ; g; Q) be a compact quaternion-Kahler manifold with
a compatible almost complex structure J . If the cohomology class [ J ] = 0 2
4
1
2
H (M; Z) then = 0.
2
Proof. If the Chern class c1(J ?) = [ 21 J ] vanishes then, by proposition 3.5, J ?
admits a nowhere vanishing section and Q admits a global admissible basis - i.e.
there exists a global compatible almost hypercomplex structure. Then we can apply
theorem 3.6. Theorem 3.8. Let (M 4n ; g; Q) be a compact quaternion-Kahler manifold with > 0.
Assume that it is not homothetic to the standard Grassmannian G2 (C m ). Then there
is no compatible almost complex structure J on (M 4n ; g; Q) .
Proof. By a result of C. LeBrun and S. Salamon [LS] we know that a compact quaternion-Kahler manifold (M 4n ; g; Q) with > 0, dierent from the complex Grassmannian G2 (C m ), has second Betti number b2(M ) = 0 . Since furthermore M must
be simply connected , we conclude that also H 2 (M; Z) = 0 [LS, proof of 0.2 p. 123].
The result follows now from Proposition 3.7. Remark 3.9. At present we do not know if there exists any compatible almost complex
structure J on G2 (C m ). M
Now we prove a fundamental identity between the Chern class c1(J ?) and the rst
Chern class c1(J ) of the tangent bundle TM 4n endowed with the compatible almost
complex structure J .
We rst remark that the curvature tensors R, R0 of the connections r, r0 are
related by
R0XY = RXY
? 21 (!2 ^ !3 )(X; Y )J ? 21 [(d!2 + !3 ^ !1)(X; Y )J2 + (d!3 + !1 ^ !2)(X; Y )J3 ]
that is, by fundamental identities (2.2),
X
R0XY = RXY + 2 F(X; Y )J + 12 d!1(X; Y )J
(3.3)
Let now recall that the rst Chern form 1 of the almost complex structure J (on
TM ) with respect to the connection r0 is given by (see for ex. [GBNV])
21(X; Y ) = ? 21 Trace(R0XY J )
(3.4)
Theorem 3.10. The Chern 2-forms for J ?, (TM; J ) respectively are related by
1 = n 21 1
(3.5)
ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS
Hence
13
c1(J ) = nc1(J ?)
(3.6)
X 0
hRXY Es; JEsi
21(X; Y ) = 12
(3.7)
Proof. By (3.4) we have
s=1;::;4n
where X; Y 2 TpM; p 2 M 4n and fE1; E2; : : : ; E4ng is an orthonormal basis of Tp M .
By using (3.3) we have
X
hR0X;Y Es; JEsi =
s=1;::;4n
+ 2
X
X
=1;2;3 s=1;::;4n
hR0X;Y Es; JEsi =
s=1;::;4n
hRX;Y Es; JEsi
s=1;::;4n
F(X; Y )hJ Es; JEsi + 12
that is
X
X
X
X
s=1;::;4n
d!1(X; Y )(JEs ; JEs)
hRXY Es; JEsi +2nF (X; Y )+2nd!1(X; Y ) (3.8)
s=1;::;4n
On the other hand we get the following. By curvature identity,
X
hRX;Y EsJEsi = ?
s=1;::;4n
X
hRJEs ;X Es; Y i ?
s=1;::;4n
X
hRY;JEs Es; X i
s=1;::;4n
Moreover, by using the known identity for the curvature tensor of a quaternionKahler manifold (see for ex [Be, pag. 403]; also [M, pag. 423]),
hRJZ T U; V i = ?hRZJT U; V i + (?hJ3U; V ihJ2 Z; T i + hJ2U; V ihJ3 Z; T i)
where U; V; Z; T 2 TpM 4n , we get
X
hRX;Y EsJEsi
s=1;::;4n
=
+
X
hREs JX Es; Y i + (hJ3Es; Y ihJ2 Es; X i ? hJ2Es; Y ihJ3 Es; X i)
s=1;::;4n
X
hRJY Es Es; X i) ? (h?J3 Es; X ihJ2 Y; Esi + hJ2Es; X ihJ3 Y; Esi)
s=1;::;4n
and, by taking into account the identity
hRUV Z; T i = hRJUJV JZ; JT i
14
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
(see [M, pag. 423]), it results
X
hRXY Es; JEsi
s=1;::;4n
=
=
X
hREs ;JX Es; Y i + 2 hY; JX i +
s=1;::;4n
X
s=1;::;4n
s=1;::;4n
X
hREs JX Es; Y i ?
Hence
X
hRJY Es Es; X i) ? 2 hX; JY i
s=1;::;4n
hRY;JEs JEs; JX i + 4 hJX; Y i
hRX;Y Es; JEsi = ?2Ric(Y; JX ) + 4 hJX; Y i
s=1;::;4n
that is
X
= ?2(n + 2) hY; JX i + 4 hJX; Y i
X
hRX;Y Es; JEsi = ?2n hJX; Y i
s=1;::;4n
(3.9)
Hence, by substitution in (3.8), we get
X
hR0X;Y Es; JEsi = ?2nF (X; Y ) + 2nF (X; Y ) + 2nd!1 (X; Y )
s=1;::;4n
= 2nd!1(X; Y )
and, nally,
that is (3.5) and (3.6) hold. 21 = nd!1
4. Compatible complex structures
A quaternion-Kahler manifold M 4n locally admits many compatible almost complex structures which are integrable, see [S3, pag. 130] and [AMP]. Any such structure
J denes a local section J : U M 4n ?! Z , U 3 x 7! Jx 2 Z of the twistor bration t : Z ?! M 4n such that the image X = J (U ) is a complex submanifold of Z .
Conversely, any complex submanifold X Z such that the projection t : X ?! M
is a dieomorphism denes a compatible complex structure on U = t(X ) M 4n .
Now we rene some of the formulas of x2 in the special case when J = J1 is an
integrable almost complex structure on the quaternion-Kahler manifold (M 4n ; g; Q) .
Let us rst recall the following result.
Lemma 4.1 ([AMP]). A compatible almost complex structure J on (M 4n ; g; Q) is
integrable if and only if
!2 J 2 = !3 J 3
for an admissible basis (J = J1; J2 ; J3) with connection 1-forms (!1 ; !2; !3 ).
ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS
15
Proposition 4.2. Assume that the compatible almost complex structure J on M
with Kahler form F and Lee form is integrable. Then one has
krF k2 = 2nkk2 ; kdF k2 = 6nkk2 ; kF k2 = kk2
and
(4.1)
kk2 + 2 = 8n
(4.2)
4 = ? 21 d(kk2 ) + d
(4.3)
Proof. The three identities (4.1) and (4.2) follow from (2.5), (2.7), (2.9) and from
(2.10) respectively, by considering that for J = J1 one has = 2!2 J2 = 2!3 J3.
The (4.3) is obtained by dierentiating (4.2). As a consequence of our previous results we deduce the following theorem obtained
in [P2] by twistor methods.
Theorem 4.3. On a compact quaternion-Kahler manifold (M 4n ; g; Q) every compatible complex structure is necessarily parallel.
Proof. If the scalar curvature is non-positive, by integrating (4.2) on M 4n we get
= = 0 and rJ = 0 by rst of (4.1).
Otherwise > 0, by Corollary 3.3 the existence of a compatible almost complex structure J implies that (M 4n ; g) is homothetic to the Grassmannian G2 (C m )
equipped with the symmetric metric. By a strong result which was proved in [BGMR,
theorem 1.7], a complex structure which is Hermitian with respect to a symmetric
metric is necessarily parallel. Therefore J is a compatible parallel complex structure
on G2(C m ). But this is impossible by theorem 2.4. Remark 4.4. If < 0 but M is non-compact the conclusion is not true: it is sucient
to think of the hyperbolic quaternionic space H Hn [P2]. M
The following two Propositions hold on any (not necessarily complete) quaternionKahler manifold.
Proposition 4.5. Let (M 4n ; g; Q) be a quaternion-Kahler manifold with a compatible complex structure J and Lee form . Then the following identities hold for any
vector elds X; Y on M 4n :
(rX )(Y ) + (rJ2Y )(J2 X ) = 12 (JX )(JY ) + 12 (J3 X )(J3 Y ) ? 2g(X; Y ) (4:4)
(rX )(Y ) + (rJ3Y )(J3 X ) = 21 (JX )(JY ) + 12 (J2 X )(J2 Y ) ? 2g(X; Y ) (4:5)
(rX )(Y ) + 21 (X )(Y ) = (rJX )(JY ) + 12 (JX )(JY )
(4:6)
16
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
Proof. From lemma 4.1 we have = 2!2 J2 ; !2 = ? 12 J2 ; !3 = ? 21 J3.
For = 2 the identity (2.2) gives
(rX !2 )(Y ) ? (rY !2)(X ) + !3 (X )!1 (Y ) ? !3(Y )!1 (X ) = g(X; J2Y )
that is
? 21 (rX )(J2 Y ) ? 21 ((rX J2)Y ) + 12 (rY )(J2 X ) + 21 ((rY J2)X )
? 21 (J3 X )!1 (Y ) + 12 (J3 Y )!1 (X )
= g(X; J2Y )
that is
? 21 (rX )(J2 Y ) ? 12 (!1 (X )J3 Y ? !3(X )JY ) + 21 (rY )(J2 X )
+ 21 (!1 (Y )J3X ? !3 (Y )JX ) ? 21 (J3 X )!1 (Y ) + 12 (J3 Y )!1 (X ) = g(X; J2Y )
that is, after suitable cancellations,
? 12 (rX )(J2 Y ) + 12 (rY )(J2 X ) + 12 !3 (X )(JY ) ? 12 !3 (Y )(JX ) = g(X; J2Y )
By changing Y with J2Y one nds
1 (r )(Y ) + 1 (r )(J X ) + 1 ! (X )(J Y ) ? 1 ! (J Y )(JX ) = ?g(X; Y )
3
2 X
2 J2 Y 2
2 3
2 3 2
and (4.4) follows by expressing !3 by . For = 3 we get
(rX !3 )(Y ) ? (rY !3)(X ) + !1 (X )!2 (Y ) ? !1(Y )!2 (X ) = g(X; J3Y )
that is equivalent to
? (rX )(J3 Y ) ? (!2 (X )JY ? !1 (X )J2 Y ) + (rY )(J3 X )
? (!2 (Y )JX ? !1(Y )J2X ) ? !1 (X )(J2 Y ) + !1(Y )(J2 X ) = 2g(X; J3Y )
and, after cancellations,
2g(X; J3Y ) = ?(rX )(J3 Y ) + (rY )(J3 X ) + 21 (J2 X )(JY ) ? 21 (J2 Y )(JY )
After substitution of Y with J3Y we get (4.5). The (4.6) follows by substraction of
(4.5) to (4.4) and substitution of X; Y with J2X; J2Y . For any point p 2 M 4n let be the orthogonal projection from the space of
bilinear forms dened on TpM 4n onto the subspace of Q-Hermitian bilinear forms,
that is for any ! 2 Bilp one has
X
! = 41 [! + !( ; )]
ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS
17
Proposition 4.6. For a compatible complex structure J with associated Lee form on a quaternion-Kahler manifold (M n ; g; Q) one has:
1) The bilinear form r + is J -Hermitian.
2) The Q-Hermitian part r = r and the skew-Q-Hermitian part r? =
(1 ? )r of the covariant derivative r are given by
(4:7)
r = 21 ( ) ? g + 21 d
4
+
1
2
+
r? = (r)s ? 21 ( ) + g
where (r)s is the symmetric part of r.
(4:8)
3) The 2-form d is Q-Hermitian and harmonic.
Proof. Statement 1) is equivalent to the identity (4.6). To prove identity (4.7) , we
calculate
X
4r+(X; Y ) 4r(X; Y ) = (rX )(Y ) + (rJX )(J Y )
using identities (4.4), (4.5), (4.6) to epress (rJ X )(J Y ), = 1; 2; 3.
The (4.8) and the rst statement of 3) follow from (4.7). To prove the last statement in 3) we can argue as in [MS] (where Q-Hermitian 2-forms are called self-dual
and it was proved that such a 2-form is closed if and only if it is harmonic): for a
Q-Hermitian 2-form ! one has ?! = c2! ^ n?1 where c2 = ?1=(2n ? 1)! (see [GP]).
Hence d = ? ? d ? d = ?c2 ? d(d ^ n?1) = 0 and 4d = 0. Remark 4.7.
(1) When J is a compatible complex structure more can be said about the 2-form
J considered in the previous section x2:
J = ? 1 ( J2) ^ ( J3) ? F
(4.9)
4
and
(4.10)
J (; J ) = ? 41 [( J2) ( J2) + ( J3) ( J3)] ? g
Hence, if > 0 then ?
J tames J , that is
g0 := 41 [( J2) ( J2) + ( J3) ( J3)] + g
(4.11)
is a Riemannian metric which is almost-Kahler with respect to J and in fact,
by a classical result, g0 is Kahler and J is parallel with respect to rg .
(2) On a non-compact complete quaternion-Kahler manifold (M 4n ; g; Q) , a complex structure compatible with Q may exist. For example, any quaternionKahler manifold (M 4n ; g; Q) which admits a simply transitive solvable group
G of isometries and dierent from quaternion hyperbolic space has a Ginvariant compatible complex structure, see [A].
0
M
18
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
Conformally balanced complex structures, Dirac and Twistor operators.
Referring to the fundamental work of Salamon [S1], we would now like to establish a
relation between compatible almost complex structures on a quaternion-Kahler manifold (M 4n ; g; Q) and two rst order dierential operators acting on sections of Q
which we will call Dirac and Twistor opertors by analogy with the four-dimensional
case; as usual they are dened by rst taking covariant derivative with respect to the
Levi-Civita connection and then projecting onto the two irreducible Sp(n) Sp(1)submodules of the target space.
Our notations are taken from the four-dimensional case and are dierent from
those used by Salamon who denotes the complexied decomposition (4.12) by S 2H E H = (E H ) (E S 3H ); what we called Twistor operator is denoted by
Salamon by D and our Dirac operator by .
Salamon also introduced complex analytic methods in quaternionic geometry by
dening the twistor bration t : Z ! M . The total space Z is a (2n +1)-dimensional
complex manifold equipped with a holomorphic contact structure and the bers
t?1 (p) are holomorphically imbedded C P1's
The general principle of Penrose is that holomorphic properties of Z reect geometric properties of (M 4n ; g; Q) . With respect to this, Salamon shows that the
kernel of the twistor operator is isomorphic to the real part of the space of holomorphic sections of the contact line bundle O(2) over Z [S1, Lemma 6.4]. He also proves
that in the compact non-Ricci-at case the Dirac operator maps the kernel of the
twistor operator isomorphically onto the space of Killing vector elds of (M; g).
In what follows we will let V denote the tangent bundle TM . Then using notations
of [AM2] we have:
where
(V Q)0
V Q = V(1)
= fA = X( J ) J ; 2 V g V(1)
=V
(4.12)
and (V Q)0 are the trace-free tensors.
P
More precisely, given a tensor T 2 V Q we can write T = J and set
X
X
= ? 31 ( J) and T 0 = ( J) J 2 V(1)
Then the above direct sum decomposition is T = T 0 + (T ? T 0 ).
Finally if p1 and p2 denote the projections onto the irreducible components we
will call
)
D : ?(Q) ! ?(V(1)
the Dirac operator given by covariant dierentiation r followed by p1 and
D : ?(Q) ! ?((V Q)0 )
ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS
19
the Twistor operator p2 r.
A non zero section of Q , restricted to the open submanifold M^ = f 6= 0g
can always be written as = fJ for some smooth function f and compatible almost
complex structure J on M . To compute the covariant derivative of we choose a
local admissible basis (J) of Q such that J = J1. We have
r = rfJ = f (!3 J2 ? !2 J3) + df J
and
3D = [f (!3 J3 + !2 J2) + df ] J1
+ [f (!3 ? !2 J1) ? df J3] J2
+ [f (?!3 J1 ? !2 ) + df J2] J3
X
= ? [f + df ] J J J
Which can also be rewritten as
3D = ?
X
(fF ) J J = ?
X
J J
(4.13)
(4:14)
so that D can be identied with the codierential : ?(Q) ?! T M .
We now compute the twistor operator:
= f!3 J2 ? f!2 J3 + df J ? D
D
(4.15)
We will now use the above facts to prove the following result which holds whether
M is compact or not, and generalizes [Theorem 2.1, P1] to higher dimensions.
Denition 4.8. A Hermitian metric (g; J ) is said to be balanced if (J is integrable
and) the Lee form = 0; conformally balanced if is exact.
For the sake of simplicity, when g is xed we also say that J is balanced or,
respectively, conformally balanced.
Of course when n = 1 one has = 0 if and only if F is symplectic. For n > 1,
by rst of (4.1) it follows immediately that a compatible complex structure J on the
quaternion-Kahler manifold (M 4n ; g; Q) is balanced if and only if it is parallel.
Theorem 4.9. Let (M 4n ; g; Q) be quaternion-Kahler if n 2 or anti-self-dual if
n = 1 with twistor space Z . Suppose J is a compatible almost complex structure and
let X denote the image of the sections J : M ! Z in the twistor space. Then,
X is associated to the holomorphic contact line bundle O(2) if and only if (g; J ) is
conformally balanced.
Proof. The smooth submanifold X given by the (disjoint) union of J (M ) and ?J (M )
is a divisor of O(2) if and only if there is a holomorphic section s 2 H 0(Z; O(2)) vanishing exactly on X . By Salamon correspondence [S1, Lemma 6.4] this is equivalent
20
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
(s) = 0 2
to having a smooth section (s) 2 ?(Q) satisfying the Twistor equation D
?(V Q)0 . Now, Salamon correspondence tells us that (s) = fJ where J is the
given almost complex structure and f is a smooth real function on M which never
vanishes because X contains no twistor bers, for example. By equations (4.13) and
= 0 is equivalent to !2 J2 = !3 J3 - i.e. J is an
(4.15) we easily see that DfJ
integrable complex structure - and furthermore d log f 2 = !2 J2 + !3 J3 - where
= !2 J2 + !3 J3 is the Lee form of (g; J ). It is now easy to check that this is
equivalent to the fact that the Hermitian metric (jf jg; J ) is conformally balanced. Remark 4.10. We can now give a more self-contained proof of the statement that
there is no compatible complex structure on a complete quaternion-Kahler manifold
(M 4n ; g; Q) of positive scalar curvature; see theorem 4.3.
Recall that M is necessarily compact and simply connected [S1]. By contradiction,
let be the Lee form of a compatible complex structure J then d is harmonic by
proposition 4.6 3), and since M is compact d = 0. In fact, since M is simply
connected we conclude that the Lee form is exact: = dh for some smooth function
h which is globally dened on M . In this situation (4:2) becomes
kdhk2 + 2h = 8n
showing that the function h cannot have a minimum; this is a contradiction because
M is compact. M
Proposition 4.11. Assume that the compatible almost complex structure J on M
is globally conformally balanced - i.e. J is integrable and = dlogf 2 is exact. Then
df J 12 f J 12 fF is a Killing 1-form (that is the vector eld dual to df J
is a Killing vector eld). Furthermore, if = 0 then the scalar curvature = 0.
Proof. We must show that under the present hypothesis the covariant derivative of
1-form df J is skew-symmetric. For any vectors X; Y , by taking into account that
df J 12 f J , one has
rX (df J )(Y ) + rY (df J )(X )
= 21 df (X )(JY ) + 21 f (rX )(JY ) + df (!3 (X )J2 Y ? !2(X )J3 Y )
+ 21 df (Y )(JX ) + 12 f (rY )(JX ) + df (!3 (Y )J2X ? !2 (Y )J3X )
= 21 f [ 12 (X )(JY ) + 21 (Y )(JX ) + (rX )(JY ) + (rY )(JX )]
(Last equality being deduced from identities df = 12 f and !2 = ? 21 J2 , !3 =
? 21 J3). By (4.6) one gets rX (df J )(Y ) + rY (df J )(X ) = 0. Finally, = 0 if
and only if = 0: in this case (g; J ) is Kahler by (3.5) and therefore the holonomy
of g is in Sp(n) Sp(1) \ U (2n) Sp(n). ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS
21
Conformally balanced hypercomplex
structures. It is well known that the
n
quaternionic projective space H P is locally hypercomplex: local admissible hypercomplex bases H = (J) on H Pn are obtained by considering systems of quaternionic
projective coordinates. But Theorem 4.3 excludes the existence of global hypercomplex bases on H Pn (In fact a classical result of W.S. Massey states that there does'nt
exist any almost complex structure on H Pn). On the other hand, as recalled by Remark 4.4, the hyperbolic quaternionic space H Hn admits global hypercomplex bases.
In fact we wonder if for < 0 this is essentially the only possibility of complete
quaternion-Kahler manifolds having this property, up to a Riemannian covering. A
rst result in this direction is the following one.
Proposition 4.12. Let (M 4n ; g; Q) be a complete quaternion-Kahler manifold. Assume that there exists a global admissible basis H = (J) of Q, consisting of integrable
complex structures. Then
1) 0 and < 0 only if (M 4n ; g) is non-compact.
Moreover
2) if < 0 and one of the complex structures J ( = 1; 2; 3) is conformally
balanced then there exists an (eventually singular) integrable distribution D on M 4n
which is Q-invariant and whose regular orbit is a (non trivial) totally geodesic quaternionic submanifold with constant quaternionic curvature, that is locally isometric to
H Hk , 1 k n, with a standard metric.
Proof. Let H = (J) be a global admissible basis which is hypercomplex. Then
by theorems 4.3 and 2.4 we exclude that 6= 0 if M 4n is compact. It remains to
prove the second statement. Hence let assume < 0, M 4n non-compact and, say J1
conformally balanced. Then formula (2.9) and similar formulas for J2; J3 together
with Lemma 4.1 imply that the three complex structures J have equal Lee forms :
= 1 = 2 = 3
and, moreover, does not vanishes.
By 1) of Proposition 4.6 the 2-tensor r + 21 is Q-Hermitian. In fact, by
summing up the (4.6) for J = J, = 1; 2; 3 respectively, one gets
r? = ? 21 + 21 ( )
and by adding to (4.7) it results, since d = 0,
(4.16)
r = ? 21 + ( ) ? g
By assumption, = dh for some function h. We dene the 1-form := e 21 h . Then
r = e 21 h [r + 12 ( )]
and hence r is a symmetric Q-Hermitian 2-tensor on M , that is g?1 is a quaternionic non-isometric innitesimal transformation on M 4n (see [AM1]). By result of
[AM3, Proposition 6] we can conclude. 22
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
5. Laplacians of self-dual forms
Let (M 4n ; g; Q) be a quaternion-Kahler manifold if n > 1 and a self-dual Einstein
4-manifold if n = 1. Then the following formulas hold.
Proposition 5.1. Let H = (J) be an admissible basis of Q. Then
4F1 = (4 + k!2k2 + k!3k2)F1
(5:1)
+ (!3? < !1; !2 >)F2 ? (!2+ < !1; !3 >)F3
and
r rF1 = (k!2 k2 + k!3k2)F1
(5:2)
+ (!3 ? < !1 ; !2 >)F2 ? (!2 + < !1 ; !3 >)F3
(Note that if > 0 then 4F1 6= 0 everywhere).
Moreover, for F = F1 one has
4F ? r rF = 4F
(5:3)
Proof. Let recall the expression of F given by (2.12). Now we compute dF . We
have
dF (X; Y ) = [rX (!3 J2 ? !2 J3)](Y ) ? [rY (!3 J2 ? !2 J3)](X )
= (rX !3)(J2 Y ) ? (rY !3 )(J2X ) ? (rX !2 )(J3Y ) + (rY !2)(J3 X )
+ !3[(!1 (X )J3 ? !3(X )J1 )(Y )] ? !2[(!2 (X )J1 ? !1 (X )J2 )(Y )]
? !3[(!1 (Y )J3 ? !3(Y )J1 )(X )] + !2[(!2 (Y )J1 ? !1(Y )J2)(X )]
Hence
dF (X; Y ) = (rX !3)(J2 Y ) ? (rY !3)(J2 X ) ? (rX !2)(J3 Y ) + (rY !2)(J3 X )
? [!2(X )!2 (J1Y ) + !3(X )!3 (J1 Y ) ? !2(Y )!2(J1 X ) ? !3(Y )!3 (J1X )]
+ !1(X )!2 (J2 Y ) ? !1(Y )!2(J2 X ) + !1(X )!3 (J3Y ) ? !1 (Y )!3 (J3X )
Let recall the expression of dF given by (2.1.2). Now we compute dF . We have
dF (X; Y ) = ?
=?
X
i
X
i
(rEi dF )(Ei ; X; Y )
[(rEi !3)(Ei )F2 (X; Y ) + !3(Ei )g((rEi J2)X; Y )
? (rEi !2)(Ei )F3 (X; Y ) ? !2 (Ei)g((rEi J3)X; Y )]
? !3(X )!1 (J3 Y ) + !3(X )!3 (J1 Y ) ? (rJ2Y !3)(X )
+ !2(X )!2 (J1 Y ) ? !2(X )!1 (J2 Y ) + (rJ3Y !3)(X )
X
? [g((rEi J2)Ei ; X )!3 (Y ) + F2 (Ei; X )(rEi !3 )(Y )
i
? g((rEi J3)Ei ; X )!2 (Y ) ? F3 (Ei; X )(rEi !2 )(Y )]
ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS
23
Hence
dF (X; Y ) = ?
X
i
[(rEi !3 )(Ei )F2 (X; Y ) ? (rEi !2 )(Ei )F3(X; Y )
+ !3 (Ei)g(!1 (Ei )J3X ? !3 (Ei)J1 X; Y )
? !2 (Ei)g(!2 (Ei )J1X ? !1 (Ei)J2 X; Y )
+ !3 (Y )g(!1 (Ei)J3 Ei ? !3(Ei )J1Ei ; X )
? !2 (Y )g(!2 (Ei)J1 Ei ? !1(Ei )J2Ei ; X )]
+ (rJ2 X !3)(Y ) ? (rJ3X !2)(Y )
? !3 (X )!1 (J3Y ) + !3(X )!3 (J1 Y ) ? (rJ2Y !3)(X )
+ !2 (X )!2 (J1Y ) ? !2(X )!1 (J2 Y ) + (rJ3Y !3)(X )
That is
dF (X; Y ) = !3F2 (X; Y ) ? !2F3 (X; Y )
? < !3; !1 > F3(X; Y ) + k!3 k2F1(X; Y )
+ k!2 k2F1(X; Y )? < !1; !2 > F2(X; Y )
+ !1 (J3X )!3 (Y ) ? !3(J1X )!3 (Y ) ? !2(J1 X )!2(Y ) + !1(J2 X )!2 (Y )
+ (rJ2 X !3)(Y ) ? (rJ3X !2 )(Y )
? !3 (X )!1 (J3Y ) + !3(X )!3 (J1 Y ) ? (rJ2Y !3)(X )
+ !2 (X )!2 (J1Y ) ? !2(X )!1 (J2 Y ) + (rJ3Y !3)(X )
By previous formulas one has
4F1(X; Y ) = (d + d)F1 (X; Y )
= (rX !3)(J2 Y ) ? (rY !3 )(J2X ) ? (rX !2)(J3 Y ) + (rY !2 )(J3X )
+ (rJ2 X !3)(Y ) ? (rJ3X !2 )(Y ) ? (rJ2Y !3 )(X ) + (rJ2Y !3)(X )
+ !1 (J3X )!3 (Y ) ? !3(J1X )!3 (Y ) ? !2(J1X )!2 (Y ) + !1(J2 X )!2 (Y )
? !3 (X )!1 (J3Y ) + !3(X )!3 (J1 Y ) + !2(X )!2 (J1 Y ) ? !2(X )!1 (J2Y )
? !2 (X )!2 (J1Y ) ? !3(X )!3 (J1 Y ) + !2(Y )!2(J1 X ) + !3(Y )!3(J1 X )
+ !1 (X )!2 (J2Y ) ? !1(Y )!2 (J2X ) + !1(X )!3 (J3 Y ) ? !1(Y )!3(J3 X )
+ (k!2 k2 + k!3k2)F1 (X; Y )
+ (!3 ? < !1; !2 >)F2(X; Y ) ? (!2 + < !1 ; !3 >)F3 (X; Y )
24
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
that is
4F1(X; Y ) = d!3(X; J2 Y ) + d!3(J2 X; Y ) ? d!2(J3X; Y ) ? d!2(X; J3 Y )
+ !1 ^ !2(J2X; Y ) + !1 ^ !2(X; J2 Y )
? !3 ^ !1(J3X; Y ) ? !3 ^ !1(X; J3 Y )
+ (k!2 k2 + k!3k2)F1 (X; Y )
+ (!3 ? < !1; !2 >)F2(X; Y ) ? (!2 + < !1; !3 >)F3(X; Y )
= g(J2X; J3 Y ) + g(X; J3J2Y ) ? g(J3X; J2 Y ) ? g(X; J2J3Y )
+ (k!2 k2 + k!3k2)F1 (X; Y )
+ (!3 ? < !1; !2 >)F2(X; Y ) ? (!2 + < !1; !3 >)F3(X; Y )
= 4g(J1X; Y )
+ (k!2 k2 + k!3k2)F1 (X; Y )
+ (!3 ? < !1; !2 >)F2(X; Y ) ? (!2 + < !1; !3 >)F3(X; Y )
and (5.1) follows immediately. Now we compute rrF1. By using (2.1.1) we have
(r rF1)(X; Y ) = ?
X
i
[(rEi !3)(Ei ))F2 (X; Y ) + !3(Ei )(rEi F2)(X; Y )
? (rEi !2 )(Ei ))F3 (X; Y ) ? !2(Ei )(rEi F3)(X; Y )]
= !3F2 (X; Y ) ? !2F3 (X; Y )
? !3 (Ei)g(!1 (Ei )J3X ? !3(Ei )J1X; Y )
+ !2(Ei )g(!2 (Ei)J1 X ? !1(Ei )J2X; Y )
and (5.2) follows. By (5.1), (5.2) it is straightforward to deduce (5.3). Now we want to compute the Laplacian on self-dual 2-forms. Let be a section of
2
+ g Q, that is is a self-dual 2-form on M. Let write = fF where F F1 is the
Kahler form of a compatible almost complex structure J J1 and H = (J1; J2; J3) is
a local basis of Q. Now we calculate the Laplacian 4. Let (Ei)i=1;::;n be an (locally
dened) orthonormal frame and X a vector eld on M . We have
X
X
(fF )(X ) = ? df (Ei )F (Ei ; X ) ? f (rEi F )(Ei ; X )
Xi
Xi
= df (Ei )g(Ei ; JX ) ? f (rEi F )(Ei ; X )
i
Hence
i
(fF ) = (df J ) + fF
Remark 5.2. Note that (fF ) = 0 () J = d(?logf ). M
(5.4)
ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS
25
Moreover, for any two vector elds X; Y we have
d(fF )(X; Y ) = rX (df J )(Y ) ?rY (df J )(X )+ df ^ F (X; Y )+ fdF (X; Y ) (5.5)
that is
d(fF ) = d(df J ) + df ^ F + fdF
(5.6)
On the other hand, we have
d d(fF ) = df ^ F + fdF
and hence
X
d(fF )(X; Y ) = ? rEi (df ^ F )(Ei ; X; Y )
Xi
? df (Ei )dF (Ei ; X; Y ) + fF (X; Y )
i
that is
X
d(fF )(X; Y ) = ? rEi (df ^ F )(Ei ; X; Y )
Xi
? df (Ei )(!3 ^ F2 ? !2 ^ F3)(Ei ; X; Y ) + fF (X; Y )
i
Now we have
X
i
(df ^ F )(Ei ; X; Y ) =
X
i
[df (Ei )F (X; Y ) + df (X )F (Y; Ei ) + df (Y )F (Ei ; X )]
and
?
X
i
rEi (df ^ F )(Ei ; X; Y ) = (df )F (X; Y ) ?
X
i
df (Ei )(rEi F )(X; Y )
? (rEi df )(X )F (Y; Ei ) ? df (X )rEi F (Y; Ei)
? (rEi df )(Y )F (Ei ; X ) ? df (Y )(rEi F )(Ei ; X )
26
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
hence
?
X
i
rEi (df ^ F )(Ei ; X; Y )
X
df (Ei ) !3 (Ei)F2 (X; Y ) ? !2 (Ei)F3 (X; Y )
X
!3(Ei )F2 (Y; Ei) ? !2(Ei )(F3 (Y; Ei)
? (rJY df )(X ) ? df (X )
i
X
!3(Ei )F2 (Ei ; X ) ? !2 (Ei)F3 (Ei ; X )
+ (rJX df )(Y ) ? df (Y )
= (df )F (X; Y ) ?
i
i
= (df )F (X; Y ) ? hdf; !3iF2 (X; Y ) + hdf; !2 iF3(X; Y )
+ (rJX df )(Y ) ? (rJY df )(X ) ? df (X )!3 (J2Y ) ? df (X )!2 (J3 Y )
+ df (Y )!3 (J2X ) ? df (Y )!2(J3 X )
= (df )F (X; Y ) ? hdf; !3iF2 (X; Y ) + hdf; !2 iF3(X; Y )
+ (rJX df )(Y ) ? (rJY df )(X ) ? df (X )(JY ) + df (Y )(JX )
Moreover we have
?
X
i
X
i
df (Ei )(!3 ^ F2 ? !2 ^ F3)(Ei ; X; Y )
[?df (Ei)!3 (Ei )F2 (X; Y ) ? df (Ei )!3 (X )F2 (Y; Ei) ? df (Ei )!3 (Y )F2 (Ei; X )
+ df (Ei )!2 (Ei )F3 (X; Y ) + df (Ei )!2 (X )F3 (Y; Ei ) + df (Ei )!2 (Y )F3 (Ei; X )]
= ?hdf; !3iF2 (X; Y ) ? (df J2)(Y )!3(X ) + (df J2)(X )!3 (Y )
+ hdf; !2 iF3 (X; Y ) + (df J3)(Y )!2 (X ) ? (df J3)(X )!2 (Y )
Hence
d(fF )(X; Y ) = (df )F (X; Y ) ? hdf; !3iF2 (X; Y ) + hdf; !2 iF3 (X; Y )
+ (rJX df )(Y ) ? (rJY df )(X ) ? df (X )( J )(Y ) + df (Y )( J )(X )
? hdf; !3 iF2 (X; Y ) ? (df J2)(Y )!3 (X ) + (df J2)(X )!3 (Y )
+ hdf; !2 iF3 (X; Y ) + (df J3)(Y )!2 (X ) ? (df J3)(X )!2 (Y )
+ fdF (X; Y )
(5.7)
ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS
27
>From which we deduce that
4(X; Y ) 4(fF )(X; Y )
= (rX df )(JY ) ? rY (df )(JX ) + (rJX df )(Y ) ? (rJY df )(X )
+ df !3 (X )J2 Y ? !2 (X )J3 Y ? df !3 (Y )J2X ? !2 (Y )J3 X
? df (J2 Y )!3 (X ) + df (J2 X )!3 (Y ) + df (J3Y )!2 (X ) ? df (J3 X )!2 (Y )
? hdf; !3 iF2(X; Y ) + hdf; !2iF3 (X; Y )
+ 4fF (X; Y ) + f 4 F (X; Y )
that is
4(fF )(X; Y ) = 4fF (X; Y ) + f 4 F (X; Y )
+ (rX df )(JY ) ? (rY df )(JX ) + (rJX df )(Y ) ? (rJY df )(X )
? hdf; !3iF2 (X; Y ) + hdf; !2iF3 (X; Y )
that is
4(fF ) = (4f )F + f 4 F ? hdf; !3 iF2 + hdf; !2iF3
(5.8)
4(fF ) = [f (4 + k!2k2 + k!3 k2) + 4f ]F
[f (!3 ? h!1 ; !2i) ? hdf; !3i]F2
+ [?f (!2 + h!1; !3 i) + hdf; !2i]F3
(5.9)
and hence, by combining with formula (5.1), we have the following
Proposition 5.3. For any self-dual 2-form = fF where F is the locally dened
Kahler form of a compatible J one has
We conclude this paragraph by showing that for Laplacian of self-dual 2-forms
associated to integrable almost complex structures, on a not necessarily compact M ,
one has the following
Proposition 5.4. Let assume that the compatible almost complex structure J on M
is integrable. Then one has
4F = (4 + 21 kk2 )F
(5.10)
and
rrF = 21 kk2 F
(5.11)
Proof. It is an immediate consequence of previous identities (5.1) and (5.2) and of
the following Lemma. 28
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
Lemma 5.5. For J integrable the following identities hold:
1
Proof. We know that:
!2+ < !1; !3 >= 0
!3? < !1; !2 >= 0
(5.12)
(5.13)
(d! + ! ^ ! )(; J ) = ?g
( = 1; 2; 3)
Hence, in particular, the 2-forms at left member are Q-Hermitian and for = 3 one
has the identities
(rX !3 )(J3Y ) ? (rJ3Y !3 )(X ) + !1(X )!2 (J3Y ) ? !1(J3 Y )!2 (X )
= (rJX !3 )(J3JY ) ? (rJ3 JY !3)(J X )
( = 1; 2; 3)
+ !1 (JX )!2 (J3JY ) ? !1(J3 JY )!2(JX )
and, by taking into account the identity
rX !3 = ?(rX !2 ) J1 + !2(X )(!2 J3) + !2 (J1X )(!2 J2)
(5.14)
which is obtained by dierentiating the identity !2 J2 = !3 J3, it results
(rX !2)(J2 Y ) ? !2 (X )!2 (Y ) + !2(J1X )!2 (J1 Y )
+ (rJ3 Y !2)(J1 X ) ? !2(J3Y )!2 (J3X ) + !2(J2Y )!2 (J2X )
+ !1 (X )!2 (J3Y ) ? !1 (J3Y )!2 (X )
= (rJX !2)(J2 JY ) ? !2 (JX )!2 (JY ) + !2(J1 JX )!2 (J1 JY )
+ (rJ3 JY !2)(J1 JX ) ? !2(J3 JY )!2 (J3JX ) + !2(J2JY )!2 (J2JX )
+ !1 (JX )!2 (J3JY ) ? !1(J3 JY )!2 (JX )
( = 1; 2; 3)
By choosing = 1 and by substituting J2Y to Y we get the identity
? (rX !2 )(Y ) ? (rJ1Y !2)(J1 X ) ? (rJ1X !2 )(J1Y ) ? (rY !2)(X )
= !2 (X )!2 (J2Y ) ? !2 (J1X )!2 (J3Y ) + !1(X )!2 (J1 Y ) ? !1(J1Y )!2 (X )
? !2(J1Y )!2 (J3X ) + !2(Y )!2(J2 X )
? !2(J1X )!2 (J3 Y ) + !2(X )!2 (J2 Y ) + !2(Y )!2(J2 X )
? !2(J1Y )!2 (J3X ) ? !1(J1 X )!2(Y ) + !1(Y )!2(J1 X )
By contraction with respect to g one obtains
4!2 = ? < !1 ; !3 > ? < !1; !3 > ? < !1 ; !3 > ? < !1; !3 >
that is
!2+ < !1; !3 >= 0
By interchanging J3 and J2 one gets also
!3? < !1; !2 >= 0
(It is sucient to repeat the computations for the admissible basis H 0 = (J0 ) where
(J1 0 = J1; J20 = J3; J30 = ?J2) and to take into account that !1 0 = !1 , !20 = !3 ,
!3 0 = ?!2. ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS
29
6. Action of the real and quaternionic
Weyl tensors on self-dual forms
Let (M 4n ; g; Q), n > 1, be a quaternion-Kahler manifold with curvature tensor R.
We have the formulas (see for example [AM2])
(
R = RH P + W Q ; = 4n(Kn+2)
R = 4n(4Kn?1) RS(1) + W R
(6.1)
where RS(1) is the curvature tensor of the 4n-dimensional sphere of constant curvature
1, RH P is the curvature tensor of the metric of quaternionic curvature 1 for the ndimensional quaternionic projective space H Pn. We recall that for any vector elds
X; Y; Z; T 2 M 4n
RS(1)(X; Y; Z; T ) = g(X; T )g(Y; Z ) ? g(X; Z )g(Y; T )
and
X
RH P (X; Y; Z; T ) = 41 [g(X; T )g(Y; Z ) ? g(X; Z )g(Y; T ) + 2 g(X; J Y )g(JZ; T )
X
X
? g(Y; JZ )g(JX; T ) + g(X; JZ )g(JY; T )]
(6.2)
W R is the Weyl curvature tensor, W Q is the quaternionic Weyl curvature tensor.
Hence
W R = 4n(nK+ 2) RH P + W Q ? 4n(4K
(6.3)
n ? 1) RS(1)
By dening the action of curvature tensor on a 2-form F by
X
(6.4)
R(F )hk = ? 21 R(Ei ; Ej ; Eh ; Ek )F (Ei ; Ej )
i;j
one has the following results.
Proposition 6.1. Let F be a self-dual 2-form on the quaternion-Kahler manifold
(M; g; Q), n > 1. Then one has
W Q (F ) = 0 ; RS(1)(F ) = F ; RH P (F ) = nF
(6.5)
W R (F ) = cF
(6.6)
? 1)(2n + 1) ]g
c = f 2Kn [ ((nn +
2)(4n ? 1)
(6.7)
and
where c is constant,
30
ALEKSEEVSKY - MARCHIAFAVA - PONTECORVO
Proof. Without any loss of generality we can assume that F is the Kahler form of a
(local) compatible almost complex structure J on (M; g; Q). We have
?2W Q (F )hk =
=
X
< W Q (Ei ; Ej )Eh ; Ek > g(IEi; Ej )
i;j
X
< W Q (Eh ; Ek )Ei ; Ej > g(IEi; Ej )
i;j
X
< W Q (Eh ; Ek )Ei ; JEi >
i
X
= ? < JW Q(Eh ; Ek )Ei; Ei >
=
i
= Trace[JW Q (Eh ; Ek )] = 0
as it was proved in [AM2]. We have
?2RS(1)(F )hk =
=
X
[g(Ei ; Ek )g(Ej ; Eh ) ? g(Ei; Eh )g(Ej ; Ek )]g(JEi; Ej )
i;j
X
i
[g(Ei ; Ek )g(JEi; Eh ) ? g(Ei; Eh )g(JEi; Ek )]
= 2g(Eh ; JEk )
= ?2Fhk
We have
?2RH P (F )hk = 41 [?2RS(1)(F )hk ]
+ 41 [2
X
g(Ei; JEj )g(JEh ; Ek ) ?
X
g(Ej ; JEh )g(JEi ; Ek )
i;j ;
X i;j;
+ g(Ei; JEh )g(JEj ; Ek )]g(JEi; Ej )
i;j ;
X
= 41 [?2F (Eh; Ek ) ? 2 g(JEj ; JEj )g(J Eh; Ek )
j ;
X
X
? g(JEi; JEh )g(JEi; Ek ) ? g(JEj ; J Eh)g(JEj ; Ek )]
i;
= 41 [?2F (Eh; Ek ) ? 8ng(JEh; Ek )
?
X
g(JEk ; JJEh ) ?
X
j ;
g(JEk ; JJEh )]
= 41 [?2F (Eh; Ek ) ? 8nF (Eh; Ek ) + 2F (Eh ; Ek )]
= ?2nF (Eh; Ek )
ALMOST COMPLEX STRUCTURES ON QUATERNION-KA HLER MANIFOLDS
Hence,
31
W R(F ) = [ 4n(nK+ 2) (n) + 0 ? 4n(4K
n ? 1) 1]F
? 1)(2n + 1) ]F
= 2Kn [ ((nn +
2)(4n ? 1)
Remark 6.2. From (6.6) it is possible to show how the formulas (2.11), (4.2) and
(5.3) could be obtained respectively from formulas (12), (26) and (7) of [G1]. Our
formulas agree with Gauduchon formulas, by taking into account that for the scalar
product between 2-forms (; ) used in [G1] one has
M
[AGS]
[A]
[ABM]
[AM1]
[AM2]
[AM3]
[AMP]
[Be]
[B]
[BGMR]
[G1]
[G2]
[G3]
[GBNV]
(F; F ) = 21 kF k2
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Gen. Antonova 2 - 99, 117279 Moscow, Russian Federation
E-mail address : [email protected]
Dipartimento di Matematica, Universita di Roma \La Sapienza", P.le A. Moro 2,
00185 Roma, Italy
E-mail address : [email protected]
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