Vanishing Theorems on Compact Hyper-kähler Manifolds

Abstract

We prove that if is a -positive holomorphic line bundle on a compact hyper-kähler manifold , then for with a nonnegative integer. In a special case, and , we recover a vanishing theorem of Verbitsky’s with a little stronger assumption.

1. Introduction

A hyper-kähler manifold is an oriented -dimensional Riemannian manifold with a special holonomy group . The holonomy group of a Kähler manifold is and the unitary group is exactly the subgroup of that preserves a complex structure, which together with a compatible Riemannian metric defines a symplectic form. Hence a Kähler manifold can also be defined as a Riemannian manifold with compatible symplectic and complex structure. By the same reasoning, is a subgroup exactly preserving three complex structures , , with . As the name suggests, a hyper-kähler manifold is also characterized as a Riemannian manifold with three compatible complex structures , , with and a compatible symplectic form which is Kähler with respect to each one of , , .

A hyper-kähler manifold is called irreducible if and . By Bogomolov’s decomposition theorem for a Kähler manifold with trivial canonical class ([1, 2]), any hyper-kähler manifold is biholomorphic to a product of irreducible hyper-kähler manifolds and a hyper-kähler complex torus up to finite cover. On any hyper-kähler manifold there is a symmetric bilinear form , called Beauville-Bogomolov-Fujiki form, which takes positive values on the Kähler cone of . The closure of the dual Kähler cone is denoted by . In [3], Verbitsky established the following vanishing theorem for a compact irreducible hyper-kähler manifold.

Theorem 1 (Verbitsky, 2007, [3]). Let be a compact irreducible hyper-kähler manifold of real dimension , and let be a holomorphic line bundle on . If , in particular, if is a positive line bundle, then

Verbitsky’s proof of the above theorem is a clever use of the symmetric pole of the complex structures and the holomorphic Bochner-Kodaira-Nakano-type identity, which appeared already in [5]. In the proof he used assumption of irreducibility of the hyper-kähler manifold. In this paper we use a different method to establish some vanishing theorems for more general hypercomplex Kähler manifolds. To get the flavor, we state the following result.

Theorem 2. Let be a compact hyper-kähler manifold of real dimension , and let be a holomorphic line bundle on . If is a positive line bundle, then for any

Note that we donot assume that is irreducible. We recover Theorem 1 if while a stronger assumption. During the proof of Theorem 2, the Kähler metric of is changed; the new Kähler metric is not necessarily hyper-kähler. So we develop our theory on hypercomplex Kähler manifolds and deal with hyper-kähler manifolds as their special examples. After deriving the Bochner-Kodaira-Nakano identities from Section 2 to Section 4, we get our main results finally in Section 5.

2. Preliminary

A Hermitian manifold is a complex manifold with an integrable complex structure and a Riemannian metric satisfying the compatible condition for any . It is called a Kähler manifold if in addition the 2-form defined by is symplectic form. is also called the Kähler form associated with and is called a Kähler metric. On the other hand, if we start with a symplectic manifold equipped with a symplectic form , then is a Kähler manifold if and only if there is a -compatible integrable complex structure. Recall that a complex structure is called -compatible if is -invariant   for any and -tamed   if . To show that two definitions of Kähler manifolds are equivalent, it suffices to note that is a Hermitian metric in the latter definition.

Definition 3. A Riemannian manifold with Riemannian metric is called a hyper-kähler manifold if it admits three integrable complex structures , , with such that is a Kähler metric with respect to each one of , , . We called a hyper-kähler metric.

Proposition 4. Let be a Kähler manifold with Levi-Civita connection . Then is hyper-kähler if there are integrable complex structures , with and , satisfying the following conditions:
(i) ;
(ii) , are parallel: .

Proof. Let ; then is a Kähler metric by the assumptions. is Hermitian relative to if , which is . Let . Clearly is nondegenerate. It is well known that if . Hence is Kähler with respect to if (i) and (ii) are true. is Kähler with respect to follows in the same way if (i) and (ii) are true.

Proposition 5. Let be a complex symplectic manifold with an integrable complex structure and an -invariant symplectic structure . Let for any . Then is a Lorentz Hermitian metric and for any point there exists a local holomorphic coordinate around such that

Proof. If moreover is -partible, then is a Käler manifold and there is a proof in [6] for this special case. The general case in our proposition follows in the same way. Since is a symplectic form, is nondegenerate but not necessarily positive definite, hence a Lorentz Hermitian metric. We could find local holomorphic coordinates at such that ; in other words, we could write locally Let us make a holomorphic change of coordinates Then in the new coordinate we have Choose . Since and are real, we have Since , in particular , From (8) and (9) we have therefore, locally .

A complex manifold with integrable complex structures , , is called a hypercomplex manifold if , and is called a hypercomplex structure (Verbitsky has studied hypercomplex manifolds and hypercomplex Kähler manifolds in a series of papers [3, 7, 8]). Obata proved that on a hypercomplex manifold there exists a unique torsion-free connection such that , , are parallel [9]: Such a connection is called an Obata connection. If is a hyper-kähler manifold, clearly the Levi-Civita connection is exactly the Obata connection of the underlying hypercomplex manifold.

The following proposition is cited from [8].

Proposition 6. Let be a hypercomplex manifold. At any point , there exists a holomorphic (with respect to ) local coordinate around with such that where , , and are the constant complex structures.

Proof. We could choose a normal coordinate at with for the Obata connection ; then the Christoffel symbols of vanish at . Since at with for example.

3. Bochner-Kodaira-Nakano-Type Identities

3.1. Generalized Hodge Identities for Differential Forms

Let be a compact hypercomplex manifold of real dimension and a Kähler manifold with Kähler metric . There are naturally associated three nondegenerate -forms: Let be a real unit orthogonal coframe of the cotangential bundle at a fixed point . Then using the Darboux theorem, we can write the Kähler form locally as Accordingly, Choose holomorphic coframes relative to the complex structure , with antiholomorphic coframes Then we have the following pointwise action at : Using holomorphic and antiholomorphic coframes, we could rewrite the 2-forms , , locally as Thus is a holomorphic -form with respect to the complex structure .

For convenience, when talking about holomorphic structure of , we always mean that it is relative to the complex structure if without special mention in the rest of this paper, though , , and have symmetric and equal roles.

Theorem 7. There exists a local holomorphic coordinate around such that

Proof. From [10], we know that there exists a local holomorphic coordinate system with respect to the complex structure such that its action is local constant: with , which coincides with the pointwise action we considered in (20). By Proposition 5, there exists a local holomorphic coordinate system such that (25) holds. By Proposition 6 and (21), (22), From (28) and the definitions of and , we conclude (26) and (27).

Let be the de Rham differential operator on , and let . Note that the complex structures , , on the tangent bundle naturally induce operator actions on the vector fields and the differential forms. Take for an example. For , the action of on differential forms is defined by The Dolbeault operators , and , are related by Clearly the Dolbeault operators , are determined completely by and the complex structure : Accordingly, the complex structures and also induce complex differential operators: where and are similar notions. Like , , , it is easy to check that the operators , , , also satisfy the graded Leibniz rule: for any , where is the degree of .

For each ordered set of indices , denote the index length by ; we write and denote by the complementary of so that where takes value if is an even permutation and otherwise. The Hodge star operator   is given by where is a function and the signature factor Given two -forms their pointwise inner product is defined by Since is compact, we can consider the Hermitian inner product on each defined by

For each , let be the wedge operator defined by and are similarly given by Let and be the adjoints of and with respect to the inner product (40), respectively. They are called contraction operators. Then for any , , If , The three equations above reflect how to commute the actions of wedges and contractions. The following Proposition 8 gives the commutation relations between contract actions and complex structure actions on differential forms. Based on them, it is easy to get the commutation relations between the actions of wedges and complex structures.

Proposition 8. Acting on the differential forms as linear operators, the contractions and the complex structures , , satisfy the commuting relations

By definition in (32) and the commutation relations in Proposition 8, we have the following expressions of differential operators via contraction and wedge operators: Let be the operators from to defined by the wedge with the 2-forms , respectively, and their adjoint operators:

The following identities in Lemma 9 are called Hodge identities [6]; they play fundamental roles in Kähler geometry. Their proof is reduced from an arbitrary Kähler manifold to the Euclidean Kähler plane via using Proposition 5. The main observation is that any intrinsically defined identity that involves the Kähler metric together with its first derivatives and which is valid for the Euclidean metric is also valid on a Kähler manifold, since by Proposition 5, a Kähler metric is oscalate order 2 to the Euclidean metric everywhere.

Lemma 9 (Hodge identities). For a proof of this lemma, please refer to [6, pages 111–114].

Proposition 10. Let be a compact hypercomplex manifold such that is a Kähler manifold; then

Proof. The idea of the proof is the same as that of Lemma 9; since by Theorem 7, the 2-forms and are oscalate order 2 to the constant 2-forms everywhere. The proof reduces to the Euclidean Kähler plane. We follow the same lines of the proof of Lemma 9 as in [6, pages 111–114].
Note every equation in the right column follows by taking conjugate of the equation in the same row of the left column, so it suffices to establish the equations in one column. Here we only give a proof of the left equation of (54). The rest of equations are proved in the same way.
By (50), we have Note that and if , we have Therefore It is not difficult to check that hence From (65),(67) we get Note that the right hand side of (68) is the conjugation of the second equation of (48); we arrive at the first equation of (54).

3.2. Twisted Bochner-Kodaira-Nakano-Type Identities

In this section we will consider the differential operators acting on bundle-valued differential forms. Suppose that is a compact hypercomplex manifold and is a Kähler manifold with Kähler metric . Given a holomorphic vector bundle over with Hermitian metric , there exists a unique connection , called Chern connection, which is compatible with the metric and satisfying . Here and are its components.

Let be a holomorphic frame of . For any -valued differential forms and of , we define their local inner product and global inner product Denote the adjoint operators of , with respect to the inner product by , . Let be the composition of and , and denote the adjoint operator of with respect to by . The operators , , , , , are defined in the same way. The operators , , , , , ,, extend naturally to . The proof of the following commuting relations among these operators acting on follows from Proposition 10.

Proposition 11.

The Chern curvature tensor of is an -valued differential form defined by The holomorphic Laplacian and antiholomorphic Laplacian are related by the classical Bochner-Kodaira-Nakano identity [11]: it plays a fundamental role in establishing many important vanishing theorems. Our naive motivation is to get more vanishing theorems by using the chances given by the other two complex structures and for a hypercomplex manifold. To this aim we define the following self-adjoint operators: Let , be the curvature components corresponding to the operators and , respectively. We have the following twisted Bochner-Kodaira-Nakano-type identities.

Proposition 12.

Proof. By Proposition 11, The second equation follows in the same way.

Recall that ; correspondingly, we define , and . Then , , and are their adjoint operators, respectively.

Proposition 13.

Proof. By Proposition 11, The rest of equations are proved in the same way.

Let be the curvature component corresponding to the operator . Let be another twisted Laplacian operator. By using Proposition 13, it is easy to prove the following Bochner-Kodaira-Nakano-type identity.

Proposition 14.

4. Local Expressions of Bochner-Kodaira-Nakano Identities

Let be a compact hypercomplex manifold and suppose that is a Kähler manifold with Kähler metric . Given a holomorphic vector bundle of rank over with Hermitian metric , let be the Chern connection of with . Write where is the connection matrix. By the compatible condition of the connection and the metric , we have by comparing the type, we get it follows that From , we know that and the Chern curvature is given by Let be the local holomorphic coordinate of such that the holomorphic coframes in (18) are represented by . Let be a holomorphic frame and the dual frame of . Let and be the Hermitian metrics on and on , respectively, and their inverses denoted, respectively, by and . Then the connection and curvature could be expressed by with where

Proposition 15. Consider

Proof. For any , Therefore, Since , we have ; hence Thus From (94) and (96), we conclude the first equation of (92). From (21) and (22), clearly . Since by definition we have .

From (21), (22), and (90), we have the following local expressions of connection and curvature: Using (91), we could write the curvature components of simply as Therefore,

The proof of the following lemma is simple via using the commuting relations (43), (45), and (44); we omit it here for brevity.

Lemma 16. Let the operators , be the wedge operators defined as in (41),(42) and , their adjoint operators; then for any integer ,

For any -valued differential form , write where the lengths and , and

Proposition 17.

Proof. Note that and if . Since and , Proposition 17 follows.

The formula in the following proposition appeared in Section 4 of [11] without proof. (Note that the expression is a little different since it uses unit orthogonal frame in [11] while here .) We will give a very simple proof here.

Proposition 18. Consider

Proof. By (89) and (90), recall that ; hence By (102) of Lemma 16, Taking inner products of both sides of the above equation with , and using (106), we get immediately (107).

Proposition 19. Consider

Proof. Using the expression (100) of the curvature , we have recall that, by (51), By (101) of Lemma 16, for any integers , we have Therefore By (103) of Lemma 16, if neither nor takes values , we have hence Take for an example; here it means in the summation of (115) we add only a restricted condition , . In others words, is a summation whose terms are the same as in (115), where indices , , , , , , vary with the same range as in (115) except that , .
By (103) of Lemma 16, for , we have hence By the same reasons, Since by (103) of Lemma 16, , we have Note that (119) could be rewritten as we remark where in the first summation of right hand side of (122) we have changed the index to (note that , , have equal positions since all of them vary form to ). Similarly, (120) could be rewritten as Combining (123), (122), and (121), we get from (117) that If we rearrange the summation range of indices , , we get Taking inner products of both sides of (125) with , and using (106), we get immediately (111).

From the definitions of , , and , we have Recall that by Proposition 15, , . By the proof of Proposition 19, in particular, from (114), we have and similarly By Propositions 12 and 14, (127) and (128), we get the following:

Proposition 20. Consider

5. Vanishing Theorems for Hypercomplex Kähler Manifolds

Let be a compact hypercomplex Kähler manifold with Kähler metric and Kähler form . The ideal to establish vanishing theorems for -valued Dolbeault cohomology groups via using the Bochner-Kodaira-Nakano identities is very simple. We use the first formula of Proposition 20, as an example. If is an arbitrary -valued -form, then an integration by part of the formula and noting that yield If , then is -harmonic and hence by the Hodge theory. Furthermore if is positive definite everywhere on , then . Hence . Thus we get a vanishing cohomology group. Therefore, using (132) to prove a vanishing theorem for -valued Dolbeault’s cohomology groups, the key point is to find conditions under which the operator is positive definite.

We can see from above reasoning, the second and third formulae of Proposition 20 produce no new vanishing theorems since their left hand sides are the same up to a positive constant. For a hypercomplex Kähler manifold , the three complex structures , , have symmetric roles; however, they are not independent of each other and related by . This may account that only two Bochner-Kodaira-Nakano identities, the formula (75) and one formula of Proposition 20, produce different vanishing theorems. Note however, that the computations of the proof (though we donot give its proof) of the last formula of Proposition 20 are simpler than the other two equations.

By (89), the Chern curvature form of is given by The first Chern class is a cohomology class which has a representation via using the Chern curvature form Conversely, any -form representing the first Chern class is in fact the Chern curvature form of some Hermitian metric on (up to a constant). In local coordinate we have with . In particular, if is a line bundle, then its curvature form represents its first Chern class up to a constant .

In [13], we introduce the following notion for semipositive holomorphic vector bundles. Based on it and the formula (75) for Kähler manifolds, we get some new vanishing theorems.

Definition 21. A holomorphic vector bundle of rank with Hermitian metric on a compact complex manifold of complex dimension is called -positive for if the following holds for any : for any -tuple vectors , , where (resp., ), the Hermitian form defined on is semipositive and the dimension of its kernel is at most , where (resp., ).

Clearly the -positivity is equivalent to the Demailly -positivity [11] and the Nakano positivity [14] is equivalent to the -positivity if . The -positivity is equivalent to the Griffiths positivity. For general integer , the positivity is a semipositive version of the Griffiths positivity [6]. A holomorphic vector bundle of arbitrary rank is called Griffiths k-positive if it is -positive.

Theorem 22. Let be a compact hypercomplex Kähler manifold of dimension and let be a Hermitian holomorphic vector bundle of rank on such that is -positive. Then(i), for and ;(ii)if in addition , then for and any nonnegative integer ,

Proof. (i) follows from Theorem 3.9 of [13] together with the fact that the anticanonical bundle of is trivial. It suffices to prove (ii). For any -valued -form , where is a fixed index. By Definition 5.1, the Hermitian form is semipositive on if ; we could diagonalize it in some local orthogonal frames such that . Here are nonnegative and for a fixed , without loss of generality, we assume that with ; in particular since . Put . Then is a positive number. Note that in the present situation the first two terms cancel each other in the first summation and the last summation vanishes in the three big summations of the Bochner-Kodaira-Nakano formula (111). Hence Thus is positive definite on -valued -forms. So we have for any and .

A holomorphic line bundle on is called -positive if there is a Hermitian metric on such that its first Chern class is semipositive and has at least positive eigenvalues [14, 15]. If is a holomorphic line bundle (denoted it by for the difference), then in Definition 21 only -positivity is applicable for , and clearly is -positive (or Griffiths -positive) if and only if it is -positive, since the first Chern class has a representation by its Chern curvature form up to a positive constant.

Theorem 23. Let be a -positive holomorphic line bundle on a compact hypercomplex Kähler manifold . Then(i), for ;(ii), for and any nonnegative integer .

Proof. We get (i) by using the Gigante-Girbau vanishing theorem on Kähler manifolds [15]; a simple proof is given in Theorem 2.4 of [13]. (i) is proved via using (75) and changing the Kähler metric on . (ii) is proved in the same way as (i) by using the Bochner-Kodaira-Nakano identities in Proposition 20. Here we give a proof of (ii) in the following paragraph.
Choose a holomorphic local coordinate system at each point , which diagonalizes simultaneously the Hermitian form and since both of them are semipositive, such that Without loss of generality, assume that . Then for any -form , the last big summation vanishes in the formula (111). is expressed by the first two big summations in (111) in the following way: Observe that if the ratio varies small when varies, for example, in the extreme case when all are equal; then is positive when . This observation tells us that if we choose the Kähler metric properly such that the eigenvalues of vary mildly relative to , then we can deduce the positivity of .
Since is -positive if and only if its curvature is semi-positive with rank , we may take a special choice of the Kähler metric of with for some positive number . Then is a compact hypercomplex manifold and is still Kähler manifold with the new Kähler form . From now on we consider as a new hypercomplex Kähler manifold with Kähler metric . Correspondingly, we have three new nondegenerate 2-froms, , , . Let , , be the associated adjoint operators of multiplication by . We could get the Bochner-Kodaira-Nakano identities with respect to , , as in Section 4. Then the eigenvalues of relative to are with Fix a point and assume that . Then . Thus for . If we choose , then for all . If , then and Since is compact, we can use a finite cover by open sets, such that, on each open set, if is a sufficiently small positive number, we may have on each open neighborhood and hence everywhere on the following: Therefore if , then is positive on and (ii) of Theorem 23 follows.

Corollary 24. Let be a holomorphic -positive line bundle on a compact hyper-kähler manifold . Then(i), for ;(ii), for and any nonnegative integer .

In particular, if is a positive holomorphic line bundle, we have, for ;, for and any nonnegative integer .

In [13], we proved that, on a compact Kähler manifold, any -ample line bundle is -positive. So Corollary 24 is also applicable to -ample line bundle. In particular, if , we get Theorem 2 for algebraic hyper-kähler manifolds.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

While the first draft of this paper was finished, the authors gave a talk at the Institute of Mathematics of Chinese University of Hong Kong, where Professor Naichung Conan Leung pointed out to the author that partial results of Theorem 1 could also be proved by using the holomorphic hard Lefschetz isomorphism (twisted by a positive line bundle) together with the Akizuki-Nakano vanishing theorem [16]. His ideal may also be applicable to other parts of our work and will be considered in another paper. He would like to thank him for his skillful comments. This work was partly supported by the Fundamental Research Funds for the Central Universities (no. 09lgpy49) and NSFC (no. 10801082).