#### 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,