# Vanishing Theorems on Compact Hyper-kähler Manifolds

Authors: Qilin Yang
Journal: Geometry
 (2014)

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## Abstract

We prove that ifB is a k-positive holomorphic line bundle on a compact hyper-kähler manifold M, then HpM,Ωq⊗B=0 for P>n+[k/2] with q a nonnegative integer. In a special case, k=0 and q=0, we recover a vanishing theorem of Verbitsky’s with a little stronger assumption.


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## Body

## 1. Introduction

A hyper-kähler manifold is an oriented4n-dimensional Riemannian manifold with a special holonomy group Sp(n)⊂SO(4n). The holonomy group of a Kähler manifold is U(n)⊂SO(2n) and the unitary group U(n) is exactly the subgroup of SO(2n) 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, Sp(n)⊂SU(2n) is a subgroup exactly preserving three complex structures I, J, K with IJ=-JI=K. As the name suggests, a hyper-kähler manifold is also characterized as a Riemannian manifold with three compatible complex structures I, J, K with IJ=-JI=K and a compatible symplectic form which is Kähler with respect to each one of I, J, K.A hyper-kähler manifoldM is called irreducible if H1(M)=0 and H2,0(M)=ℂ. 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 M there is a symmetric bilinear form q, called Beauville-Bogomolov-Fujiki form, which takes positive values on the Kähler cone 𝒦 of (M,I). The closure of the dual Kähler cone 𝒦∨=:{x∈H1,1(M,ℝ)∣q(x,y)>0,∀y∈𝒦} is denoted by 𝒦∨¯. In [3], Verbitsky established the following vanishing theorem for a compact irreducible hyper-kähler manifold.Theorem 1 (Verbitsky, 2007, [3]).
LetM be a compact irreducible hyper-kähler manifold of real dimension 4n, and let L be a holomorphic line bundle on M. If 0≠c1(L)∈𝒦∨¯, in particular, if L is a positive line bundle, then
(1)Hp(M,L)=0,forp>n.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.
LetM be a compact hyper-kähler manifold of real dimension 4n, and let L be a holomorphic line bundle on (M,I). If L is a positive line bundle, then for any q≥0(2)Hp(M,Ωq⊗L)=0,forp>n.Note that we donot assume thatM is irreducible. We recover Theorem 1 if q=0 while a stronger assumption. During the proof of Theorem 2, the Kähler metric of (M,I) 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 manifoldM is a complex manifold with an integrable complex structure I and a Riemannian metric g satisfying the compatible condition g(IX,IY)=g(X,Y) for any X,Y∈TM. It is called a Kähler manifold if in addition the 2-form ω defined by ω(X,Y)=-g(X,IY) is symplectic form. ω is also called the Kähler form associated with g and g is called a Kähler metric. On the other hand, if we start with a symplectic manifold M equipped with a symplectic form ω, then M is a Kähler manifold if and only if there is a ω-compatible integrable complex structure. Recall that a complex structure I is called ω-compatible if ω is I-invariant  ω(IX,IY)=ω(X,Y) for any X,Y∈TM and I-tamed  ω(X,IX)>0 if X≠0. To show that two definitions of Kähler manifolds are equivalent, it suffices to note that g(X,Y)=ω(X,IY) is a Hermitian metric in the latter definition.Definition 3.
A Riemannian manifoldM with Riemannian metric g is called a hyper-kähler manifold if it admits three integrable complex structures I, J, K with IJ=-JI=K such that g is a Kähler metric with respect to each one of I, J, K. We called g a hyper-kähler metric.Proposition 4.
Let(M,ω,I) be a Kähler manifold with Levi-Civita connection ∇. Then M is hyper-kähler if there are integrable complex structures J, K with IJ=-JI=K and J, K satisfying the following conditions:
(i)ω(JX,KY)=ω(X,IY);
(ii)J, K are parallel: ∇J=∇K=0.Proof.
Letg(X,Y)=ω(X,IY); then g is a Kähler metric by the assumptions. g is Hermitian relative to J if g(X,Y)=g(JX,JY), which is ω(JX,KY)=ω(X,IY). Let ωJ(X,Y)=-g(X,JY)=-ω(X,KY). Clearly ωJ is nondegenerate. It is well known that dωJ=0 if ∇J=0. Hence g is Kähler with respect to J if (i) and (ii) are true. g is Kähler with respect to K follows in the same way if (i) and (ii) are true.Proposition 5.
LetM be a complex symplectic manifold with an integrable complex structure I and an I-invariant symplectic structure ω. Let g(X,Y)=ω(X,IY) for any X,Y∈TM. Then g is a Lorentz Hermitian metric and for any point x∈M there exists a local holomorphic coordinate (w1,…,wn) around x such that
(3)g=∑jk‍(±δjk+O(|w|2))dwjdw¯k.Proof.
If moreoverI is ω-partible, then M 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 ω is a symplectic form, g is nondegenerate but not necessarily positive definite, hence a Lorentz Hermitian metric. We could find local holomorphic coordinates (zj) at x such that gjk(x)=±δjk; in other words, we could write locally
(4)g=∑jkl‍(±δjk+ajklzl+ajkl¯z¯lhhh+O(|z|2)ajkl¯z¯l)dzjdz¯k;(5)ω=i∑jkl‍(±δjk+ajklzl+ajkl¯z¯lhhhh+O(|z|2)ajkl¯z¯l)dzj∧dz¯k.
Let us make a holomorphic change of coordinates
(6)zk=wk+12∑‍bklmwlwm.
Then in the new coordinate we have
(7)g=∑jk‍(±δjk+∑l‍(ajklwl+ajkl¯w¯l+bkljwlhhhhhhhhhhhhh+bjlk¯w¯l+O(|w|2))∑2003l‍)dwjdw¯k.
Choose bklj=-ajkl. Since g and ω are real, we have
(8)ajkl¯=ajkl¯.
Since dω=0, in particular dω(x)=0,
(9)ajkl=aljk.
From (8) and (9) we have
(10)bjlk¯=-akjl¯=-ajkl¯;
therefore, locally g=∑jk‍(±δjk+O(|w|2))dwjdw¯k.A complex manifoldM with integrable complex structures I, J, K is called a hypercomplex manifold if IJ=-JI=K, and (I,J,K) 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 (M,I,J,K) there exists a unique torsion-free connection such that I, J, K are parallel [9]:
(11)∇I=∇J=∇K=0.
Such a connection is called an Obata connection. If M 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(M,I,J,K) be a hypercomplex manifold. At any point x∈M, there exists a holomorphic (with respect to I) local coordinate (zj) around x with zj(x)=0 such that
(12)I(z)=I0+O(|z|2)I′;J(z)=J0+O(|z|2)J′;K(z)=K0+O(|z|2)K′,
where I0, J0, and K0 are the constant complex structures.Proof.
We could choose a normal coordinate(zj) at x with zj(x)=0 for the Obata connection ∇; then the Christoffel symbols of ∇ vanish at x. Since at x(13)∇I=dI=∇J=dJ=∇K=dK=0,I(z)=I0+O(|z|2)I′ with I0=I(0) for example.


## 3. Bochner-Kodaira-Nakano-Type Identities

### 3.1. Generalized Hodge Identities for Differential Forms

Let(M,I,J,K) be a compact hypercomplex manifold of real dimension 4n and (M,I) a Kähler manifold with Kähler metric g. There are naturally associated three nondegenerate 2-forms:
(14)ωI(·,·)=g(·,I·),ωJ(·,·)=g(·,J·),ωK(·,·)=g(·,K·).
Let {ξ1,Iξ1,Jξ1,Kξ1,…,ξn,Iξn,Jξn,Kξn} be a real unit orthogonal coframe of the cotangential bundle T*M at a fixed point x∈M. Then using the Darboux theorem, we can write the Kähler form ωI locally as
(15)ωI=∑j=1n‍(ξj∧Iξj+Jξj∧Kξj).
Accordingly,
(16)ωJ=∑j=1n‍(ξj∧Jξj+Kξj∧Iξj),(17)ωK=∑j=1n‍(ξj∧Kξj+Iξj∧Jξj).
Choose holomorphic coframes relative to the complex structure I,
(18)θj=ξj-iIξj,θj+n=Jξj-iKξj,j=1,…,n,
with antiholomorphic coframes
(19)θ¯j=ξj+iIξj,θ¯j+n=Jξj+iKξj,j=1,…,n.
Then we have the following pointwise action at x∈M:
(20)Iθj=iθj,Iθ¯j=-iθ¯j,j=1,…,2n;(21)Jθj=θ¯j+n,Jθj+n=-θ¯j,j=1,…,n;(22)Kθj=-iθ¯j+n,Kθj+n=iθ¯j,j=1,…,n.
Using holomorphic and antiholomorphic coframes, we could rewrite the 2-forms ωI, ωJ, ωK locally as
(23)ωI=i2∑j=1n‍(θj∧θ¯j+θj+n∧θ¯j+n),ωJ=12∑j=1n‍(θj∧θj+n+θ¯j∧θ¯j+n),ωK=i2∑j=1n‍(θ¯j∧θ¯j+n-θj∧θj+n).
Thus
(24)φ=ωJ+iωK=∑j=1n‍θj∧θj+n
is a holomorphic (2,0)-form with respect to the complex structure I.For convenience, when talking about holomorphic structure ofM, we always mean that it is relative to the complex structure I if without special mention in the rest of this paper, though I, J, and K have symmetric and equal roles.Theorem 7.
There exists a local holomorphic coordinate(z1,…,zn) around x such that
(25)ωI=∑jk‍(δjk+O(|z|2))dzj∧dzk;(26)ωJ=∑jk‍(δjk+O(|z|2))(dzj∧dzk+n+dz¯j∧dz¯k+n);(27)ωK=∑jk‍(δjk+O(|z|2))(dzj∧dzk+n-dz¯j∧dz¯k+n).Proof.
From [10], we know that there exists a local holomorphic coordinate system (zj) with respect to the complex structure I such that its action is local constant: Idzj=idzj with θj=dzj, which coincides with the pointwise action we considered in (20). By Proposition 5, there exists a local holomorphic coordinate system (zj) such that (25) holds. By Proposition 6 and (21), (22),
(28)Jdzj=dzj+n+O(|z|2)∑k‍(dzk+dz¯k);Jdzj+n=-dzj+O(|z|2)∑k‍(dzk+dz¯k);Kdzj=idzj+n+O(|z|2)∑k‍(dzk+dz¯k);Kdzj+n=-idzj+O(|z|2)∑k‍(dzk+dz¯k).
From (28) and the definitions of ωJ and ωK, we conclude (26) and (27).Letd:Ωp(M)→Ωp+1(M) be the de Rham differential operator on M, and let dc=I-1dI. Note that the complex structures I, J, K on the tangent bundle naturally induce operator actions on the vector fields and the differential forms. Take I for an example. For α,…,β∈Ω•(M), the action of I on differential forms is defined by
(29)I(α∧⋯∧β)=Iα∧⋯∧Iβ.
The Dolbeault operators ∂, ∂¯ and d, dc are related by
(30)d=∂+∂¯,dc=-i(∂-∂¯).
Clearly the Dolbeault operators ∂, ∂¯ are determined completely by d and the complex structure I:
(31)∂=12(d+idc);∂¯=12(d-idc).
Accordingly, the complex structures J and K also induce complex differential operators:
(32)dJ=J-1dJ=∂J+∂¯J,dK=K-1dK=∂K+∂¯K,
where ∂J=J-1∂J and ∂¯J,∂K,∂¯K are similar notions. Like d, ∂, ∂¯, it is easy to check that the operators dJ, dK, ∂J, ∂K also satisfy the graded Leibniz rule: for any ξ,η∈Ω•(M),
(33)dJ(ξ∧η)=dJξ∧η+(-1)|ξ|ξ∧dJη,
where |ξ| is the degree of ξ.For each ordered set of indicesA={α1,…,αp}, denote the index length by |A|=p; we write
(34)θA=θα1∧⋯∧θαp,θ¯A=θ¯α1∧⋯∧θ¯αp,
and denote by A^={αp+1,…,αn} the complementary of A so that
(35)θA∧θA^=(-1)τ(A)θ1∧⋯∧θ2n,
where τ(A) takes value 1 if A is an even permutation and -1 otherwise. The Hodge star operator  * is given by
(36)*(fθA∧θB¯)=2|A|+|B|-2n(-1)nϵABf¯θA^∧θB^¯,
where f is a function and the signature factor
(37)ϵAB=(-1)n(2n-1)+(2n-p)q+τ(A)+τ(B).
Given two (p,q)-forms
(38)ξ=1p!q!∑A,B‍ξAB¯θA∧θ¯B,η=1p!q!∑A,B‍ηAB¯θA∧θ¯B,
their pointwise inner product is defined by
(39)〈ξ,η〉=ξ∧*η=(1p!q!∑A,B‍ξAB¯η¯AB¯)1(2n)!ωI2n.
Since M is compact, we can consider the Hermitian inner product on each Ωp,q(M) defined by
(40)(ξ,η)=∫M‍〈ξ,η〉,ξ,η∈Ωp,q(M),For eachk=1,…,2n, let ek:Ωp,q(M)→Ωp+1,q(M) be the wedge operator defined by
(41)ek(η)=θk∧η;
and e¯k:Ωp,q(M)→Ωp,q+1(M) are similarly given by
(42)e¯k(η)=θ¯k∧η.
Let ik and i¯k be the adjoints of ek and e¯k with respect to the inner product (40), respectively. They are called contraction operators. Then for any k, l=1,…,2n,
(43)eki¯l+i¯lek=0;(44)ekik+ikek=2.
If k≠l,
(45)ekil+ilek=0.
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 contractionsik and the complex structures I, J, K satisfy the commuting relations
(46)ikI=-iIik,i¯kI=iIi¯k;ikJ=Ji¯k+n,ik+nJ=-Ji¯k;ikK=iKi¯k+n,ik+nK=-iKi¯k.By definition in (32) and the commutation relations in Proposition 8, we have the following expressions of differential operators via contraction and wedge operators:
(47)∂=∑k‍(ek∂k+ek+n∂k+n),∂*=-∑k‍(∂¯kik+∂¯k+nik+n);(48)∂J=∑k‍(e¯k+n∂k-e¯k∂k+n),∂J*=∑k‍(-∂¯ki¯k+n+∂¯k+ni¯k);(49)∂K=i∑k‍(e¯k+n∂k-e¯k∂k+n),∂K*=i∑k‍(-∂¯k+ni¯k+∂¯ki¯k+n).
Let L=LI,LJ,LK be the operators from Ω*(E) to Ω*(E) defined by the wedge with the 2-forms ω=ωI,ωJ,ωK, respectively, and Λ=ΛI,ΛJ,ΛK their adjoint operators:
(50)L=i2∑k‍(ekek+e¯k+ne¯k+n),Λ=-i2∑k‍(ikik+i¯k+ni¯k+n);(51)LJ=12∑k‍(ekek+n+e¯ke¯k+n),ΛJ=12∑k‍(ikik+n+i¯ki¯k+n);(52)LK=i2∑k‍(e¯ke¯k+n-ekek+n),ΛK=i2∑k‍(ikik+n-i¯ki¯k+n).The following identities in Lemma9 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).
(53)
[
Λ
,
∂
]
=
i
∂
¯
*
,
[
Λ
,
∂
¯
]
=
-
i
∂
*
.
For a proof of this lemma, please refer to [6, pages 111–114].Proposition 10.
Let(M,I,J,K) be a compact hypercomplex manifold such that (M,I) is a Kähler manifold; then
(54)[ΛJ,∂]=∂¯J*,[ΛJ,∂¯]=∂J*;(55)[ΛK,∂]=-∂¯K*,[ΛK,∂¯]=-∂K*;(56)[ΛI,∂J]=i∂¯J*,[ΛI,∂¯J]=-i∂J*;(57)[ΛJ,∂J]=-∂¯*,[ΛJ,∂¯J]=-∂*;(58)[ΛK,∂J]=i∂¯*,[ΛK,∂¯J]=-i∂*;(59)[ΛI,∂K]=i∂¯K*,[ΛI,∂¯K]=-i∂K*;(60)[ΛK,∂K]=∂¯*,[ΛK,∂¯K]=∂*;(61)[ΛJ,∂K]=-i∂¯*,[ΛJ,∂¯K]=i∂*.Proof.
The idea of the proof is the same as that of Lemma9; since by Theorem 7, the 2-forms ωJ and ωK 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
(62)2[ΛJ,∂]=∑k,l‍[ikik+n+i¯ki¯k+n,∂lel]=∑k‍[ikik+n+i¯ki¯k+n,∂kek+∂k+nek+n]+∑l≠k,k+n‍[ikik+n+i¯ki¯k+n,∂lel].
Note that
(63)[i¯ki¯k+n,∂kek]=[i¯ki¯k+n,∂k+nek+n]=0,
and if l≠k,k+n, we have
(64)[ikik+n,∂lel]=[i¯ki¯k+n,∂lel]=0.
Therefore
(65)2[ΛJ,∂]=∑k‍([ikik+n,∂kek]+[ikik+n,∂k+nek+n]).
It is not difficult to check that
(66)[ikik+n,ek]=-2ik+n,[ikik+n,ek+n]=2ik;
hence
(67)[ikik+n,∂kek]=∂k[ikik+n,ek]=-2∂kik+n,[ikik+n,∂k+nek+n]=∂k+n[ikik+n,ek+n]=2∂k+nik.
From (65),(67) we get
(68)[ΛJ,∂]=∑k‍(-∂kik+n+∂k+nik).
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(M,I,J,K) is a compact hypercomplex manifold and (M,I) is a Kähler manifold with Kähler metric g. Given a holomorphic vector bundle E over M with Hermitian metric h, there exists a unique connection D, called Chern connection, which is compatible with the metric h and satisfying D′′=∂¯. Here D=D′+D′′ and
(69)D′:Ωp,q(E)⟶Ωp+1,q(E),D′′:Ωp,q(E)⟶Ωp,q+1(E)
are its components.Let{sα} be a holomorphic frame of E. For any E-valued differential forms ξ=∑α‍ξα⊗sα and η=∑β‍ηα⊗sα of Ωp,q(E), we define their local inner product
(70)〈ξ,η〉=∑α,β‍hαβ¯ξα∧*ηβ
and global inner product
(71)(ξ,η)E=∫M‍〈ξ,η〉.
Denote the adjoint operators of D′, D′′ with respect to the inner product by δ′, δ′′. Let DJ′:Ωp,q(E)→Ωp,q+1(E) be the composition of
(72)J-1⊗idE:E⊗Ωp,q(M)⟶E⊗Ωq,p(M),D′ and J⊗idE, and denote the adjoint operator of DJ′ with respect to (,)E by δJ′. The operators DJ′′, δJ′′, DK′, δK′, DK′′, δK′′ are defined in the same way. The operators *, d, Λ, L, ΛJ, LJ,ΛK, LK extend naturally to Ωp,q(E). The proof of the following commuting relations among these operators acting on Ωp,q(E) follows from Proposition 10.Proposition 11.
(73)
[
Λ
,
D
′
]
=
i
δ
′′
,
[
Λ
,
D
′′
]
=
-
i
δ
′
;
[
Λ
J
,
D
′
]
=
δ
J
′′
,
[
Λ
J
,
D
′′
]
=
δ
J
′
;
[
Λ
K
,
D
′
]
=
-
δ
K
′′
,
[
Λ
K
,
D
′′
]
=
-
δ
K
′
;
[
Λ
I
,
D
J
′
]
=
i
δ
J
′′
,
[
Λ
I
,
D
J
′′
]
=
-
i
δ
J
′
;
[
Λ
J
,
D
J
′
]
=
-
δ
′′
,
[
Λ
J
,
D
J
′′
]
=
-
δ
′
;
[
Λ
K
,
D
J
′
]
=
i
δ
′′
,
[
Λ
K
,
D
J
′′
]
=
-
i
δ
′
;
[
Λ
I
,
D
K
′
]
=
i
δ
K
′′
,
[
Λ
I
,
D
K
′′
]
=
-
i
δ
K
′
;
[
Λ
K
,
D
K
′
]
=
δ
′′
,
[
Λ
K
,
D
K
′′
]
=
δ
′
;
[
Λ
J
,
D
K
′
]
=
-
i
δ
′′
,
[
Λ
J
,
D
K
′′
]
=
i
δ
′
.TheChern curvature tensor Θ of E is an End(E)-valued differential form defined by
(74)(D′′D′+D′D′′)ξ=Θ∧ξ=e(Θ)ξ∈Ωp+1,q+1(E),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhξ∈Ωp,q(E).
The holomorphic Laplacian Δ′=D′δ′+δ′D′ and antiholomorphic Laplacian Δ′′=D′′δ′′+δ′′D′′ are related by the classical Bochner-Kodaira-Nakano identity [11]:
(75)Δ′′-Δ′=[e(iΘ),Λ],
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 J and K for a hypercomplex manifold. To this aim we define the following self-adjoint operators:
(76)ΔJ′=DJ′δJ′+δJ′DJ′,ΔK′=DK′δK′+δK′DK′.
Let ΘJ, ΘK be the curvature components corresponding to the operators D′′DJ′+DJ′D′′ and D′′DK′+DK′D′′, respectively. We have the following twisted Bochner-Kodaira-Nakano-type identities.Proposition 12.
(77)
Δ
′′
-
Δ
J
′
=
[
e
(
Θ
J
)
,
Λ
J
]
,
Δ
′′
-
Δ
K
′
=
-
[
e
(
Θ
K
)
,
Λ
K
]
.Proof.
By Proposition11,
(78)Δ′′-ΔJ′=D′′δ′′+δ′′D′′-(DJ′δJ′+δJ′DJ′)=-D′′[ΛJ,DJ′]-[ΛJ,DJ′]D′′-DJ′[ΛJ,D′′]-[ΛJ,D′′]D′J=-D′′ΛJDJ′+D′′DJ′ΛJ-ΛJDJ′D′′+DJ′ΛJD′′-DJ′ΛJD′′+DJ′D′′ΛJ-ΛJD′′DJ′+D′′ΛJDJ′=(D′′DJ′+DJ′D′′)ΛJ-ΛJ(D′′DJ′+DJ′D′′)=[e(ΘJ),ΛJ].
The second equation follows in the same way.Recall thatφ=ωJ+iωK; correspondingly, we define Lφ=LJ+iLK, Dφ¯′=(1/2)(DJ′-iDK′) and Dφ¯′′=(1/2)(DJ′′-iDK′′). Then Λφ=ΛJ-iΛK, Dφ¯′=(1/2)(δJ′+iδK′), and δφ¯′′=(1/2)(δJ′′+iδK′′) are their adjoint operators, respectively.Proposition 13.
(79)
[
Λ
φ
,
D
′
]
=
δ
φ
¯
′′
,
[
Λ
φ
,
D
φ
¯
′
]
=
-
δ
′′
;
[
Λ
φ
,
D
′′
]
=
δ
φ
¯
′
,
[
Λ
φ
,
D
φ
¯
′′
]
=
-
δ
′
.Proof.
By Proposition11,
(80)[Λφ,D′]=12([ΛJ,D′]-i[ΛK,D′])=12(δJ′′+iδK′′)=δφ¯′′,(81)[Λφ,Dφ¯′]=12([ΛJ,DJ′-iDK′]-i[ΛK,DJ′-iDK′])=12([ΛJ,DJ′]+i(-[ΛJ,DK′]hhhhhhhhhhh-[ΛK,DJ′])-[ΛK,DK′])=-δ′′.
The rest of equations are proved in the same way.LetΘφ¯ be the curvature component corresponding to the operator D′′Dφ¯′+Dφ¯′D′′. Let Δφ¯′=Dφ¯′δφ¯′+δφ¯′Dφ¯′ be another twisted Laplacian operator. By using Proposition 13, it is easy to prove the following Bochner-Kodaira-Nakano-type identity.Proposition 14.
(82)
Δ
′′
-
Δ
′
φ
¯
=
[
e
(
Θ
φ
¯
)
,
Λ
φ
]
.


## 3.1. Generalized Hodge Identities for Differential Forms

Let(M,I,J,K) be a compact hypercomplex manifold of real dimension 4n and (M,I) a Kähler manifold with Kähler metric g. There are naturally associated three nondegenerate 2-forms:
(14)ωI(·,·)=g(·,I·),ωJ(·,·)=g(·,J·),ωK(·,·)=g(·,K·).
Let {ξ1,Iξ1,Jξ1,Kξ1,…,ξn,Iξn,Jξn,Kξn} be a real unit orthogonal coframe of the cotangential bundle T*M at a fixed point x∈M. Then using the Darboux theorem, we can write the Kähler form ωI locally as
(15)ωI=∑j=1n‍(ξj∧Iξj+Jξj∧Kξj).
Accordingly,
(16)ωJ=∑j=1n‍(ξj∧Jξj+Kξj∧Iξj),(17)ωK=∑j=1n‍(ξj∧Kξj+Iξj∧Jξj).
Choose holomorphic coframes relative to the complex structure I,
(18)θj=ξj-iIξj,θj+n=Jξj-iKξj,j=1,…,n,
with antiholomorphic coframes
(19)θ¯j=ξj+iIξj,θ¯j+n=Jξj+iKξj,j=1,…,n.
Then we have the following pointwise action at x∈M:
(20)Iθj=iθj,Iθ¯j=-iθ¯j,j=1,…,2n;(21)Jθj=θ¯j+n,Jθj+n=-θ¯j,j=1,…,n;(22)Kθj=-iθ¯j+n,Kθj+n=iθ¯j,j=1,…,n.
Using holomorphic and antiholomorphic coframes, we could rewrite the 2-forms ωI, ωJ, ωK locally as
(23)ωI=i2∑j=1n‍(θj∧θ¯j+θj+n∧θ¯j+n),ωJ=12∑j=1n‍(θj∧θj+n+θ¯j∧θ¯j+n),ωK=i2∑j=1n‍(θ¯j∧θ¯j+n-θj∧θj+n).
Thus
(24)φ=ωJ+iωK=∑j=1n‍θj∧θj+n
is a holomorphic (2,0)-form with respect to the complex structure I.For convenience, when talking about holomorphic structure ofM, we always mean that it is relative to the complex structure I if without special mention in the rest of this paper, though I, J, and K have symmetric and equal roles.Theorem 7.
There exists a local holomorphic coordinate(z1,…,zn) around x such that
(25)ωI=∑jk‍(δjk+O(|z|2))dzj∧dzk;(26)ωJ=∑jk‍(δjk+O(|z|2))(dzj∧dzk+n+dz¯j∧dz¯k+n);(27)ωK=∑jk‍(δjk+O(|z|2))(dzj∧dzk+n-dz¯j∧dz¯k+n).Proof.
From [10], we know that there exists a local holomorphic coordinate system (zj) with respect to the complex structure I such that its action is local constant: Idzj=idzj with θj=dzj, which coincides with the pointwise action we considered in (20). By Proposition 5, there exists a local holomorphic coordinate system (zj) such that (25) holds. By Proposition 6 and (21), (22),
(28)Jdzj=dzj+n+O(|z|2)∑k‍(dzk+dz¯k);Jdzj+n=-dzj+O(|z|2)∑k‍(dzk+dz¯k);Kdzj=idzj+n+O(|z|2)∑k‍(dzk+dz¯k);Kdzj+n=-idzj+O(|z|2)∑k‍(dzk+dz¯k).
From (28) and the definitions of ωJ and ωK, we conclude (26) and (27).Letd:Ωp(M)→Ωp+1(M) be the de Rham differential operator on M, and let dc=I-1dI. Note that the complex structures I, J, K on the tangent bundle naturally induce operator actions on the vector fields and the differential forms. Take I for an example. For α,…,β∈Ω•(M), the action of I on differential forms is defined by
(29)I(α∧⋯∧β)=Iα∧⋯∧Iβ.
The Dolbeault operators ∂, ∂¯ and d, dc are related by
(30)d=∂+∂¯,dc=-i(∂-∂¯).
Clearly the Dolbeault operators ∂, ∂¯ are determined completely by d and the complex structure I:
(31)∂=12(d+idc);∂¯=12(d-idc).
Accordingly, the complex structures J and K also induce complex differential operators:
(32)dJ=J-1dJ=∂J+∂¯J,dK=K-1dK=∂K+∂¯K,
where ∂J=J-1∂J and ∂¯J,∂K,∂¯K are similar notions. Like d, ∂, ∂¯, it is easy to check that the operators dJ, dK, ∂J, ∂K also satisfy the graded Leibniz rule: for any ξ,η∈Ω•(M),
(33)dJ(ξ∧η)=dJξ∧η+(-1)|ξ|ξ∧dJη,
where |ξ| is the degree of ξ.For each ordered set of indicesA={α1,…,αp}, denote the index length by |A|=p; we write
(34)θA=θα1∧⋯∧θαp,θ¯A=θ¯α1∧⋯∧θ¯αp,
and denote by A^={αp+1,…,αn} the complementary of A so that
(35)θA∧θA^=(-1)τ(A)θ1∧⋯∧θ2n,
where τ(A) takes value 1 if A is an even permutation and -1 otherwise. The Hodge star operator  * is given by
(36)*(fθA∧θB¯)=2|A|+|B|-2n(-1)nϵABf¯θA^∧θB^¯,
where f is a function and the signature factor
(37)ϵAB=(-1)n(2n-1)+(2n-p)q+τ(A)+τ(B).
Given two (p,q)-forms
(38)ξ=1p!q!∑A,B‍ξAB¯θA∧θ¯B,η=1p!q!∑A,B‍ηAB¯θA∧θ¯B,
their pointwise inner product is defined by
(39)〈ξ,η〉=ξ∧*η=(1p!q!∑A,B‍ξAB¯η¯AB¯)1(2n)!ωI2n.
Since M is compact, we can consider the Hermitian inner product on each Ωp,q(M) defined by
(40)(ξ,η)=∫M‍〈ξ,η〉,ξ,η∈Ωp,q(M),For eachk=1,…,2n, let ek:Ωp,q(M)→Ωp+1,q(M) be the wedge operator defined by
(41)ek(η)=θk∧η;
and e¯k:Ωp,q(M)→Ωp,q+1(M) are similarly given by
(42)e¯k(η)=θ¯k∧η.
Let ik and i¯k be the adjoints of ek and e¯k with respect to the inner product (40), respectively. They are called contraction operators. Then for any k, l=1,…,2n,
(43)eki¯l+i¯lek=0;(44)ekik+ikek=2.
If k≠l,
(45)ekil+ilek=0.
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 contractionsik and the complex structures I, J, K satisfy the commuting relations
(46)ikI=-iIik,i¯kI=iIi¯k;ikJ=Ji¯k+n,ik+nJ=-Ji¯k;ikK=iKi¯k+n,ik+nK=-iKi¯k.By definition in (32) and the commutation relations in Proposition 8, we have the following expressions of differential operators via contraction and wedge operators:
(47)∂=∑k‍(ek∂k+ek+n∂k+n),∂*=-∑k‍(∂¯kik+∂¯k+nik+n);(48)∂J=∑k‍(e¯k+n∂k-e¯k∂k+n),∂J*=∑k‍(-∂¯ki¯k+n+∂¯k+ni¯k);(49)∂K=i∑k‍(e¯k+n∂k-e¯k∂k+n),∂K*=i∑k‍(-∂¯k+ni¯k+∂¯ki¯k+n).
Let L=LI,LJ,LK be the operators from Ω*(E) to Ω*(E) defined by the wedge with the 2-forms ω=ωI,ωJ,ωK, respectively, and Λ=ΛI,ΛJ,ΛK their adjoint operators:
(50)L=i2∑k‍(ekek+e¯k+ne¯k+n),Λ=-i2∑k‍(ikik+i¯k+ni¯k+n);(51)LJ=12∑k‍(ekek+n+e¯ke¯k+n),ΛJ=12∑k‍(ikik+n+i¯ki¯k+n);(52)LK=i2∑k‍(e¯ke¯k+n-ekek+n),ΛK=i2∑k‍(ikik+n-i¯ki¯k+n).The following identities in Lemma9 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).
(53)
[
Λ
,
∂
]
=
i
∂
¯
*
,
[
Λ
,
∂
¯
]
=
-
i
∂
*
.
For a proof of this lemma, please refer to [6, pages 111–114].Proposition 10.
Let(M,I,J,K) be a compact hypercomplex manifold such that (M,I) is a Kähler manifold; then
(54)[ΛJ,∂]=∂¯J*,[ΛJ,∂¯]=∂J*;(55)[ΛK,∂]=-∂¯K*,[ΛK,∂¯]=-∂K*;(56)[ΛI,∂J]=i∂¯J*,[ΛI,∂¯J]=-i∂J*;(57)[ΛJ,∂J]=-∂¯*,[ΛJ,∂¯J]=-∂*;(58)[ΛK,∂J]=i∂¯*,[ΛK,∂¯J]=-i∂*;(59)[ΛI,∂K]=i∂¯K*,[ΛI,∂¯K]=-i∂K*;(60)[ΛK,∂K]=∂¯*,[ΛK,∂¯K]=∂*;(61)[ΛJ,∂K]=-i∂¯*,[ΛJ,∂¯K]=i∂*.Proof.
The idea of the proof is the same as that of Lemma9; since by Theorem 7, the 2-forms ωJ and ωK 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
(62)2[ΛJ,∂]=∑k,l‍[ikik+n+i¯ki¯k+n,∂lel]=∑k‍[ikik+n+i¯ki¯k+n,∂kek+∂k+nek+n]+∑l≠k,k+n‍[ikik+n+i¯ki¯k+n,∂lel].
Note that
(63)[i¯ki¯k+n,∂kek]=[i¯ki¯k+n,∂k+nek+n]=0,
and if l≠k,k+n, we have
(64)[ikik+n,∂lel]=[i¯ki¯k+n,∂lel]=0.
Therefore
(65)2[ΛJ,∂]=∑k‍([ikik+n,∂kek]+[ikik+n,∂k+nek+n]).
It is not difficult to check that
(66)[ikik+n,ek]=-2ik+n,[ikik+n,ek+n]=2ik;
hence
(67)[ikik+n,∂kek]=∂k[ikik+n,ek]=-2∂kik+n,[ikik+n,∂k+nek+n]=∂k+n[ikik+n,ek+n]=2∂k+nik.
From (65),(67) we get
(68)[ΛJ,∂]=∑k‍(-∂kik+n+∂k+nik).
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(M,I,J,K) is a compact hypercomplex manifold and (M,I) is a Kähler manifold with Kähler metric g. Given a holomorphic vector bundle E over M with Hermitian metric h, there exists a unique connection D, called Chern connection, which is compatible with the metric h and satisfying D′′=∂¯. Here D=D′+D′′ and
(69)D′:Ωp,q(E)⟶Ωp+1,q(E),D′′:Ωp,q(E)⟶Ωp,q+1(E)
are its components.Let{sα} be a holomorphic frame of E. For any E-valued differential forms ξ=∑α‍ξα⊗sα and η=∑β‍ηα⊗sα of Ωp,q(E), we define their local inner product
(70)〈ξ,η〉=∑α,β‍hαβ¯ξα∧*ηβ
and global inner product
(71)(ξ,η)E=∫M‍〈ξ,η〉.
Denote the adjoint operators of D′, D′′ with respect to the inner product by δ′, δ′′. Let DJ′:Ωp,q(E)→Ωp,q+1(E) be the composition of
(72)J-1⊗idE:E⊗Ωp,q(M)⟶E⊗Ωq,p(M),D′ and J⊗idE, and denote the adjoint operator of DJ′ with respect to (,)E by δJ′. The operators DJ′′, δJ′′, DK′, δK′, DK′′, δK′′ are defined in the same way. The operators *, d, Λ, L, ΛJ, LJ,ΛK, LK extend naturally to Ωp,q(E). The proof of the following commuting relations among these operators acting on Ωp,q(E) follows from Proposition 10.Proposition 11.
(73)
[
Λ
,
D
′
]
=
i
δ
′′
,
[
Λ
,
D
′′
]
=
-
i
δ
′
;
[
Λ
J
,
D
′
]
=
δ
J
′′
,
[
Λ
J
,
D
′′
]
=
δ
J
′
;
[
Λ
K
,
D
′
]
=
-
δ
K
′′
,
[
Λ
K
,
D
′′
]
=
-
δ
K
′
;
[
Λ
I
,
D
J
′
]
=
i
δ
J
′′
,
[
Λ
I
,
D
J
′′
]
=
-
i
δ
J
′
;
[
Λ
J
,
D
J
′
]
=
-
δ
′′
,
[
Λ
J
,
D
J
′′
]
=
-
δ
′
;
[
Λ
K
,
D
J
′
]
=
i
δ
′′
,
[
Λ
K
,
D
J
′′
]
=
-
i
δ
′
;
[
Λ
I
,
D
K
′
]
=
i
δ
K
′′
,
[
Λ
I
,
D
K
′′
]
=
-
i
δ
K
′
;
[
Λ
K
,
D
K
′
]
=
δ
′′
,
[
Λ
K
,
D
K
′′
]
=
δ
′
;
[
Λ
J
,
D
K
′
]
=
-
i
δ
′′
,
[
Λ
J
,
D
K
′′
]
=
i
δ
′
.TheChern curvature tensor Θ of E is an End(E)-valued differential form defined by
(74)(D′′D′+D′D′′)ξ=Θ∧ξ=e(Θ)ξ∈Ωp+1,q+1(E),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhξ∈Ωp,q(E).
The holomorphic Laplacian Δ′=D′δ′+δ′D′ and antiholomorphic Laplacian Δ′′=D′′δ′′+δ′′D′′ are related by the classical Bochner-Kodaira-Nakano identity [11]:
(75)Δ′′-Δ′=[e(iΘ),Λ],
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 J and K for a hypercomplex manifold. To this aim we define the following self-adjoint operators:
(76)ΔJ′=DJ′δJ′+δJ′DJ′,ΔK′=DK′δK′+δK′DK′.
Let ΘJ, ΘK be the curvature components corresponding to the operators D′′DJ′+DJ′D′′ and D′′DK′+DK′D′′, respectively. We have the following twisted Bochner-Kodaira-Nakano-type identities.Proposition 12.
(77)
Δ
′′
-
Δ
J
′
=
[
e
(
Θ
J
)
,
Λ
J
]
,
Δ
′′
-
Δ
K
′
=
-
[
e
(
Θ
K
)
,
Λ
K
]
.Proof.
By Proposition11,
(78)Δ′′-ΔJ′=D′′δ′′+δ′′D′′-(DJ′δJ′+δJ′DJ′)=-D′′[ΛJ,DJ′]-[ΛJ,DJ′]D′′-DJ′[ΛJ,D′′]-[ΛJ,D′′]D′J=-D′′ΛJDJ′+D′′DJ′ΛJ-ΛJDJ′D′′+DJ′ΛJD′′-DJ′ΛJD′′+DJ′D′′ΛJ-ΛJD′′DJ′+D′′ΛJDJ′=(D′′DJ′+DJ′D′′)ΛJ-ΛJ(D′′DJ′+DJ′D′′)=[e(ΘJ),ΛJ].
The second equation follows in the same way.Recall thatφ=ωJ+iωK; correspondingly, we define Lφ=LJ+iLK, Dφ¯′=(1/2)(DJ′-iDK′) and Dφ¯′′=(1/2)(DJ′′-iDK′′). Then Λφ=ΛJ-iΛK, Dφ¯′=(1/2)(δJ′+iδK′), and δφ¯′′=(1/2)(δJ′′+iδK′′) are their adjoint operators, respectively.Proposition 13.
(79)
[
Λ
φ
,
D
′
]
=
δ
φ
¯
′′
,
[
Λ
φ
,
D
φ
¯
′
]
=
-
δ
′′
;
[
Λ
φ
,
D
′′
]
=
δ
φ
¯
′
,
[
Λ
φ
,
D
φ
¯
′′
]
=
-
δ
′
.Proof.
By Proposition11,
(80)[Λφ,D′]=12([ΛJ,D′]-i[ΛK,D′])=12(δJ′′+iδK′′)=δφ¯′′,(81)[Λφ,Dφ¯′]=12([ΛJ,DJ′-iDK′]-i[ΛK,DJ′-iDK′])=12([ΛJ,DJ′]+i(-[ΛJ,DK′]hhhhhhhhhhh-[ΛK,DJ′])-[ΛK,DK′])=-δ′′.
The rest of equations are proved in the same way.LetΘφ¯ be the curvature component corresponding to the operator D′′Dφ¯′+Dφ¯′D′′. Let Δφ¯′=Dφ¯′δφ¯′+δφ¯′Dφ¯′ be another twisted Laplacian operator. By using Proposition 13, it is easy to prove the following Bochner-Kodaira-Nakano-type identity.Proposition 14.
(82)
Δ
′′
-
Δ
′
φ
¯
=
[
e
(
Θ
φ
¯
)
,
Λ
φ
]
.


## 4. Local Expressions of Bochner-Kodaira-Nakano Identities

Let(M,I,J,K) be a compact hypercomplex manifold and suppose that (M,I) is a Kähler manifold with Kähler metric g. Given a holomorphic vector bundle E of rank r over M with Hermitian metric h, let D=D′+D′′ be the Chern connection of E with D′′=∂¯. Write
(83)D′=∂+ϑ,
where ϑ∈Ω1,0(End(E)) is the connection matrix. By the compatible condition of the connection D and the metric h, we have
(84)dh=hϑ+ϑ¯th;
by comparing the type, we get
(85)∂h=hϑ,∂¯h=ϑ¯th;
it follows that
(86)ϑ=h-1∂h.
From ∂2=∂¯2=0, we know that
(87)∂ϑ=-ϑ∧ϑ,
and the Chern curvature Θ is given by
(88)Θ=∂¯ϑ=dϑ+ϑ∧ϑ.
Let {zj} be the local holomorphic coordinate of M such that the holomorphic coframes in (18) are represented by θj=dzj. Let {sα} be a holomorphic frame and {sα} the dual frame of E. Let (gjk¯) and (hαβ¯) be the Hermitian metrics on M and on E, respectively, and their inverses denoted, respectively, by (gij¯) and (hαβ¯). Then the connection and curvature could be expressed by
(89)ϑ=∑α,β‍ϑβαsα⊗sβ,Θ=∑α,β‍Θβαsα⊗sβ=∑α,β,j,k‍Rβjk¯αdzj∧dz¯k⊗sα⊗sβ
with
(90)ϑβα=∑γ,j‍hαγ¯∂hβγ¯∂zjdzj;Θβα=∑j,k‍Rβjk¯αdzj∧dz¯k=∑γ,j,k‍∂∂z¯k(hαγ¯hβγ¯∂zj)dz¯k∧dzj,
where
(91)Rβjk¯α=-∑γ‍hαγ¯∂j∂¯khβγ¯+∑γ,λ,μ‍hαγ¯hλμ¯∂jhβμ¯∂¯khλγ¯.Proposition 15.
Consider(92)ΘJ=∂¯ϑJ;ΘK=iΘJ;Θφ¯=ΘJ.Proof.
For anyξ=∑α‍ξα⊗sα∈Ωp,q(E),
(93)DJ′ξ=∑α‍(∂Jξα)sα+∑α,β‍(J-1ϑαβJξα)sβ=(∂J+J-1ϑJ)ξ=(∂J+ϑJ)ξ.
Therefore,
(94)DJ′D′′ξ=∑α‍(∂J∂¯ξα)sα+∑α,β‍(J-1ϑαβJ∂¯ξα)sβD′′DJ′ξ=∑α‍∂¯(∂Jξα)sα+∑α,β‍∂¯(J-1ϑαβJ)ξαsβ-∑α,β‍(J-1ϑαβJ∂¯ξα)sβ.
Since DJ=D′J=D′′J=0, we have ∂¯J=J∂¯; hence
(95)∂¯∂J=∂¯(J-1∂J)=J-1∂¯∂J,∂J∂¯=J-1∂J∂¯=J-1∂∂¯J.
Thus
(96)∂¯∂J+∂J∂¯=0.
From (94) and (96), we conclude the first equation of (92). From (21) and (22), clearly ΘK=∂¯ϑK=iΘJ. Since by definition
(97)Θφ¯=12(∂¯ϑJ-i∂¯ϑK)=12(ΘJ-iΘK),
we have Θφ¯=ΘJ.From (21), (22), and (90), we have the following local expressions of connection and curvature:
(98)(ϑJ)βα=∑j=1n‍hαγ¯∂hβγ¯∂zjdz¯j+n-∑j=1n‍hαγ¯∂hβγ¯∂zj+ndz¯j,(ΘJ)βα=∑k=12n‍∑j=1n‍∂∂z¯k(hαγ¯∂hβγ¯∂zj)dz¯k∧dz¯j+n-∑k=12n‍∑j=1n‍∂∂z¯k(hαγ¯∂hβγ¯∂zj+n)dz¯k∧dz¯j.
Using (91), we could write the curvature components of ΘJ simply as
(99)(ΘJ)βα=-∑k=12n‍∑j=1n‍Rβjk¯αdz¯k∧dz¯j+n+∑k=12n‍∑j=1n‍Rβj+nk¯αdz¯k∧dz¯j.
Therefore,
(100)ΘJ=∑k=12n‍∑j=1n(-Rβjk¯αdz¯k∧dz¯j+n⊗sα⊗sβ‍hhhhhh+Rβj+nk¯αdz¯k∧dz¯j⊗sα⊗sβ).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 operatorsek, e¯k be the wedge operators defined as in (41),(42) and ik, i¯k their adjoint operators; then for any integer 1≤p,q,k≤2n,
(101)[e¯pe¯q,ikik+n]=0;(102)[epe¯q,iki¯k]={0,p,q≠k;-2e¯qi¯k,p=k,q≠k;-2epik,q=k,p≠k;4-2ekik-2e¯ki¯k,p=q=k;(103)[e¯pe¯q,i¯ki¯k+n]={0,p,q≠k,k+n;-2e¯qi¯k+n,p=k,q≠k,k+n;2e¯pi¯k+n,q=k,p≠k,k+n;2e¯qi¯k,p=k+n,q≠k,k+n;-2e¯pi¯k,q=k+n,p≠k,k+n;4-2e¯ki¯k-2e¯k+ni¯k+n,p=k,q=k+n;-4+2e¯ki¯k+2e¯k+ni¯k+n,q=k,p=k+n.For anyE-valued differential form ξ∈Ωp,q(E), write
(104)ξ=∑α‍ξα⊗sα=∑P,Q,α‍ξP,Q,αθP∧θ¯Q⊗sα,
where the lengths |P|=p and |Q|=q, and
(105)ξα=∑P,Q‍ξP,Q,αθP∧θ¯Q∈Ωp,q(M).Proposition 17.
(106)
(
e
q
i
k
ξ
,
ξ
)
E
=
2
∑
α
,
P
,
Q
‍
∑
(
S
k
)
=
(
T
q
)
=
P
‍
ξ
S
k
,
Q
,
α
ξ
¯
T
q
,
Q
,
α
.Proof.
Note thatik(θP∧θ¯Q)=0 and ik(θk∧θP∧θ¯Q)=2θP∧θ¯Q if k∉P. Since (eqiku,u)E=(ikξ,iqξ)E and (θk∧θP∧θ¯Q,θk∧θP∧θ¯Q)E=2(θP∧θ¯Q,θP∧θ¯Q)E, 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 〈θk,θk〉=2.) We will give a very simple proof here.Proposition 18.
Consider(107)12〈[e(iΘ),Λ]ξ,ξ〉=∑α,β,P,Q‍∑(Sk¯)=(Tq¯)=Q‍Rβkq¯αξP,Sk¯,αξ¯P,Tq¯,β+∑α,β,P,Q‍∑(Sk)=(Tq)=P‍Rβpk¯αξSk,Q,αξ¯Tq,Q,β-∑α,β,k,P,Q‍Rβkk¯αξP,Q,αξ¯P,Q,β.Proof.
By (89) and (90),
(108)e(iΘ)=i∑α,β,p,q‍Rβpq¯αepe¯q⊗sα⊗sβ;
recall that Λ=(i/2)∑k‍iki¯k; hence
(109)[e(iΘ),Λ]ξ=-12∑p,q,α,β,k,P,Q‍Rβpq¯α([epe¯q,iki¯k]ξβ)sα.
By (102) of Lemma 16,
(110)[e(iΘ),Λ]ξ=∑α,β,k,P,Q‍∑q≠k‍Rβkq¯α(e¯qi¯kξβ)sα+∑α,β,k,P,Q‍∑p≠k‍Rβpk¯α(epikξβ)sα+∑α,β,k,P,Q‍Rβkk¯α(e¯ki¯kξβ)sα+∑α,β,k,P,Q‍Rβkk¯α(ekikξβ)sα-2∑α,β,k,P,Q‍Rβkk¯αξβsα=∑α,β,q,k,P,Q‍Rβkq¯α(e¯qi¯kξβ)sα+∑α,β,p,k,P,Q‍Rβpk¯α(epikξβ)sα-2∑α,β,k,P,Q‍Rβkk¯αξβsα.
Taking inner products of both sides of the above equation with ξ, and using (106), we get immediately (107).Proposition 19.
Consider(111)12〈[e(ΘJ),ΛJ]ξ,ξ〉=∑α,β,P,Q‍∑p,q=12n‍∑(Sq¯)=(Tp¯)=Q‍Rβpq¯αξP,Sq¯,αξ¯P,Tp¯,β-∑α,β,P,Q‍∑k=12n‍Rβkk¯αξP,Q,αξ¯P,Q,β+∑α,β,P,Q‍∑p,q=1n‍(∑(Sq¯)=(Tp¯)=Q‍Rβp+nq+n¯αξP,Sq¯,αξ¯P,Tp¯,βhhhhhhhhhhhhh+∑(Sp+n¯)=(Tq+n¯)=Q‍Rβpq¯αξP,Sp+n¯,αξ¯P,Tq+n¯,β)+∑α,β,P,Q‍∑p,q=1n‍(∑(Sp¯)=(Tq+n¯)=Q‍Rβpq+n¯αξP,Sp¯,αξ¯P,Tq+n¯,βhhhhhhhhhhhhhh+∑(Sp+n¯)=(Tq¯)=Q‍Rβp+nq¯αξP,Sp+n¯,αξ¯P,Tq¯,βhhhhhhhhhhhhhh-∑(Sq¯)(Tp+n¯)=Q‍Rβpq+n¯αξP,Sq¯,αξ¯P,Tp+n¯,βhhhhhhhhhhhhhh-∑(Sq+n¯)=(Tp¯)=Q‍Rβp+nq¯αξP,Sq+n¯,αξ¯P,Tp¯,β).Proof.
Using the expression (100) of the curvature ΘJ, we have
(112)e(ΘJ)=∑α,β=1r‍∑p,q=1n‍(-Rβpq¯αe¯qe¯p+n-Rβpq+¯nαe¯q+ne¯p+nhhhhhhhhhhh+Rβp+nq¯αe¯qe¯p+Rβp+nq+¯nαe¯q+ne¯p)⊗sα⊗sβ;
recall that, by (51),
(113)ΛJ=12∑k=1n‍(ikik+n+i¯ki¯k+n).
By (101) of Lemma 16, for any integers 1≤p,q,k≤2n, we have
(114)[e¯qe¯p,ikik+n]=0.
Therefore
(115)[e(ΘJ),ΛJ]ξ=12∑α,β=1r‍∑p,q,k=1n‍∑P,Q‍(-Rβpq¯α[e¯qe¯p+n,i¯ki¯k+n]ξβhhhhhhhhhhhhhhh-Rβpq+¯nα[e¯q+ne¯p+n,i¯ki¯k+n]ξβhhhhhhhhhhhhhhh+Rβp+nq¯α[e¯qe¯p,i¯ki¯k+n]ξβhhhhhhhhhhhhhhh+Rβp+nq+¯nα[e¯q+ne¯p,i¯ki¯k+n]ξβ)sα.
By (103) of Lemma 16, if neither p nor q takes values k,k+n, we have
(116)[e¯pe¯q,i¯ki¯k+n]=[e¯pe¯q,i¯ki¯k+n]=0;
hence
(117)[e(ΘJ),ΛJ]ξ=∑p=k;q≠k‍+∑p≠k;q=k‍+∑p=k;q=k‍.
Take ∑p=k;q≠k‍ for an example; here it means in the summation of (115) we add only a restricted condition p=k, q≠k. In others words, ∑p=k;q≠k‍ is a summation whose terms are the same as in (115), where indices α, β, P, Q, p, q, k vary with the same range as in (115) except that p=k, q≠k.
By (103) of Lemma 16, for j≠k,k+n, we have
(118)[e¯je¯k,i¯ki¯k+n]=2e¯ji¯k+n,[e¯je¯k+n,i¯ki¯k+n]=-2e¯ji¯k;
hence
(119)∑p=k;q≠k‍=∑α,β,P,Q,q≠k‍(Rβkq¯αe¯qi¯kξβ+Rβkq+n¯αe¯q+ni¯kξβhhhhhhhhhhhh+Rβk+nq¯αe¯qi¯k+nξβ+Rβk+nq+n¯αe¯q+ni¯k+nξβ)sα.
By the same reasons,
(120)∑p≠k;q=k‍=∑α,β,P,Q,p≠k‍(Rβpk¯αe¯p+ni¯k+nξβ-Rβpk+n¯αe¯p+ni¯kξβhhhhhhhhhhh-Rβp+nk¯αe¯pi¯k+nξβ+Rβp+nk+n¯αe¯pi¯kξβ)sα.
Since by (103) of Lemma 16, [e¯ke¯k+n,i¯ki¯k+n]=4-2e¯ki¯k-2e¯k+ni¯k+n, we have
(121)∑p=k;q=k‍=-∑α,β,P,Q‍(Rβkk¯α+Rβk+nk+n¯α)hhhhhhh×(2-e¯ki¯k-e¯k+ni¯k+n)ξβsα.
Note that (119) could be rewritten as
(122)∑p=k;q≠k‍=∑α,β,P,Q,p,q(Rβpq¯αe¯qi¯pξβ+Rβpq+n¯αe¯q+ni¯pξβ‍hhhhhhhhh+Rβp+nq¯αe¯qi¯p+nξβ+Rβp+nq+n¯αe¯q+ni¯p+nξβ)sα-∑α,β,P,Q,k‍(Rβkk¯αe¯ki¯kξβ+Rβkk+n¯αe¯k+ni¯kξβhhhhhhhhhhh+Rβk+nk¯αe¯ki¯k+nξβ+Rβk+nk+n¯αe¯k+ni¯k+nξβ)sα;
we remark where in the first summation of right hand side of (122) we have changed the index k to q (note that p, q, k have equal positions since all of them vary form 1 to n). Similarly, (120) could be rewritten as
(123)∑p≠k;q=k‍=∑α,β,P,Q,p,q(Rβpq¯αe¯p+ni¯q+nξβ-Rβpq+n¯αe¯p+ni¯qξβ‍hhhhhhhhh-Rβp+nq¯αe¯pi¯q+nξβ+Rβp+nq+n¯αe¯pi¯qξβ)sα-∑α,β,P,Q,k‍(Rβkk¯αe¯k+ni¯k+nξβ-Rβkk+n¯αe¯k+ni¯kξβhhhhhhhh-Rβk+nk¯αe¯ki¯k+nξβ+Rβk+nk+n¯αe¯ki¯kξβ)sα.
Combining (123), (122), and (121), we get from (117) that
(124)[e(ΘJ),ΛJ]ξ=∑α,β=1r‍∑p,q=1n‍∑P,Q(Rβpq¯αe¯qi¯pξβ+Rβpq+n¯αe¯q+ni¯pξβ‍hhhhhhhhhhhh+Rβp+nq¯αe¯qi¯p+nξβ+Rβp+nq+n¯αe¯q+ni¯p+nξβhhhhhhhhhhhh+Rβpq¯αe¯p+ni¯q+nξβ-Rβpq+n¯αe¯p+ni¯qξβhhhhhhhhhhhh-Rβp+nq¯αe¯pi¯q+nξβ+Rβp+nq+n¯αe¯pi¯qξβ)sα-2∑α,β=1r‍∑k=12n‍∑P,Q‍Rβkk¯αξβsα.
If we rearrange the summation range of indices p, q, we get
(125)[e(ΘJ),ΛJ]ξ=∑α,β=1r‍∑p,q=12n‍∑P,Q‍Rβpq¯αe¯qi¯pξβ-2∑α,β=1r‍∑k=12n‍∑P,Q‍Rβkk¯αξβsα+∑α,β=1r‍∑p,q=1n‍∑P,Q‍(Rβpq¯αe¯p+ni¯q+nξβ+Rβp+nq+n¯αe¯pi¯qξβ)sα+∑α,β=1r‍∑p,q=1n‍∑P,Q(Rβpq+n¯αe¯q+ni¯pξβ‍+Rβp+nq¯αe¯qi¯p+nξβhhhhhhhhhhhhhh-Rβpq+n¯αe¯p+ni¯qξβ-Rβp+nq¯αe¯pi¯q+nξβ)sα.
Taking inner products of both sides of (125) with ξ, and using (106), we get immediately (111).From the definitions ofΛJ, ΛK, and Λφ, we have
(126)Λφ=∑k=1n‍i¯ki¯k+n.
Recall that by Proposition 15, ΘK=iΘJ, Θφ¯=ΘJ. By the proof of Proposition 19, in particular, from (114), we have
(127)[e(ΘK),ΛK]=[e(iΘJ),i2∑k=1n‍i¯ki¯k+n]=-[e(ΘJ),ΛJ],
and similarly
(128)[e(Θφ¯),Λφ]=[e(ΘJ),∑k=1n‍i¯ki¯k+n]=2[e(ΘJ),ΛJ].
By Propositions 12 and 14, (127) and (128), we get the following:Proposition 20.
Consider(129)Δ′′-ΔJ′=[e(ΘJ),ΛJ];(130)Δ′′-ΔK′=[e(ΘJ),ΛJ];(131)Δ′′-Δφ¯′=2[e(ΘJ),ΛJ].


## 5. Vanishing Theorems for Hypercomplex Kähler Manifolds

Let(M,I,J,K) be a compact hypercomplex Kähler manifold with Kähler metric h and Kähler form ω=ωI. The ideal to establish vanishing theorems for E-valued Dolbeault cohomology groups via using the Bochner-Kodaira-Nakano identities is very simple. We use the first formula of Proposition 20,
(132)Δ′′-ΔJ′=[e(ΘJ),ΛJ],
as an example. If ξ is an arbitrary E-valued (p,q)-form, then an integration by part of the formula
(133)((Δ′′-ΔJ′)ξ,ξ)E=([e(ΘJ),ΛJ]ξ,ξ)E
and noting that (ΔJ′ξ,ξ)E=∥DJ′ξ∥E2+∥δJ′ξ∥E2≥0 yield
(134)∥D′′ξ∥E+∥D′′*ξ∥E≥∫M‍〈[ΘJ,ΛJ]ξ,ξ〉.
If ξ∈Hp,q(E), then ξ is Δ′′-harmonic and hence D′′ξ=D′′*ξ=0 by the Hodge theory. Furthermore if [ΘJ,ΛJ] is positive definite everywhere on Ωp,q(E), then ξ=0. Hence Hp,q(M,E)≅Hq(M,Ωp(E))≅Hp,q(E)=0. Thus we get a vanishing cohomology group. Therefore, using (132) to prove a vanishing theorem for E-valued Dolbeault’s cohomology groups, the key point is to find conditions under which the operator [ΘJ,ΛJ] is positive definite.We can see from above reasoning, the second and third formulae of Proposition20 produce no new vanishing theorems since their left hand sides are the same up to a positive constant. For a hypercomplex Kähler manifold (M,I,J,K), the three complex structures I, J, K have symmetric roles; however, they are not independent of each other and related by IJ=-JI=K. 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 E is given by
(135)Θ=Θ(E)=∑α,β‍Θβαsα⊗sβ=∑α,β,j,k‍Rβjk¯αdzj∧dz¯k⊗sα⊗sβ.
The first Chern class c1(E)∈H2(M,ℝ) is a cohomology class which has a representation via using the Chern curvature form
(136)c1(E)=12πTrE(iΘ(E))=i2πΘ(det⁡(E)).
Conversely, any 2-form representing the first Chern class c1(E) is in fact the Chern curvature form of some Hermitian metric on det⁡E (up to a constant). In local coordinate we have
(137)iΘ(det⁡⁡(E))=i∑j,k‍Rjk¯dzj∧dz¯k=-∂∂¯log⁡det⁡(hαβ¯)
with Rjk¯=∑α‍Rαjk¯α. In particular, if E is a line bundle, then its curvature form represents its first Chern class up to a constant 1/2π.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 bundleE of rank r with Hermitian metric h on a compact complex manifold M of complex dimension n is called (k,s)-positive for 1≤s≤r if the following holds for any x∈M: for any s-tuple vectors vj∈V, 1≤j≤s, where V=Ex (resp., TxM), the Hermitian form
(138)Qx(•,•)=iΘ(E)(∑j=1s‍·⊗vj,∑j=1s‍·⊗vj),hhhhhhhhhhhhhhhhhhhh•∈W⊕s,·∈W
defined on W⊕s is semipositive and the dimension of its kernel is at most k, where W=TxM (resp., Ex).Clearly the(0,s)-positivity is equivalent to the Demailly s-positivity [11] and the Nakano positivity [14] is equivalent to the (0,s)-positivity if s≥min⁡{n,r}. The (0,1)-positivity is equivalent to the Griffiths positivity. For general integer k, the (k,1) positivity is a semipositive version of the Griffiths positivity [6]. A holomorphic vector bundle E of arbitrary rank is called Griffiths k-positive if it is (k,1)-positive.Theorem 22.
LetM be a compact hypercomplex Kähler manifold of dimension 4n and let E be a Hermitian holomorphic vector bundle of rank r on M such that E is (k,s)-positive. Then(i)
Hp(M,E)=0, for p>k and s≥min{2n-p+1,r};(ii)
if in additionk≤2n-1, then for s≥min{2n-p+1,r} and any nonnegative integer p,
(139)H2n(M,Ωp⊗E)=0.Proof.
(i) follows from Theorem 3.9 of [13] together with the fact that the anticanonical bundle of M is trivial. It suffices to prove (ii). For any E-valued (p,2n)-form ξ=∑P,α‍ξP,QαθP∧θ¯1∧⋯∧θ¯2n∧⊗eα∈Ωp,2n(E), where Q=12⋯2n is a fixed index. By Definition 5.1, the Hermitian form iΘ(E) is semipositive on Ωp,2n(E) if s≥min⁡{2n-p+1,r}; we could diagonalize it in some local orthogonal frames such that Rαβ¯jk¯=λjαδαβδjk. Here (λjα)1≤j≤2n,1≤α≤r are nonnegative and for a fixed α, without loss of generality, we assume that λ1α≤λ2α≤⋯≤λ2nα with λk+1α>0; in particular λ2nα>0 since k≤2n-1. Put λ=min⁡{λ2nα∣1≤α≤r}. Then λ 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
(140)12〈[e(ΘJ),ΛJ]ξ,ξ〉=∑α,P‍∑j=1n‍(∑j∈Q‍λjα|ξP,Qα|2+∑j+n∈Q‍λj+nα|ξP,Qα|2)≥∑P‍∑α‍(λ1α+λ2α+⋯+λ2nα)|ξP,Qα|2≥λ(∑P‍∑α‍|ξP,Qα|2)=λ|ξ|2.
Thus [e(ΘJ),ΛJ] is positive definite on E-valued (p,2n)-forms. So we have H2n(M,Ωp⊗E)=0 for any p and s≥min⁡{2n-p+1,r}.A holomorphic line bundleB on M is called k-positive if there is a Hermitian metric on B such that its first Chern class c1(B) is semipositive and has at least n-k positive eigenvalues [14, 15]. If E is a holomorphic line bundle (denoted it by B for the difference), then in Definition 21 only (k,1)-positivity is applicable for B, and clearly B is (k,1)-positive (or Griffiths k-positive) if and only if it is k-positive, since the first Chern class has a representation by its Chern curvature form up to a positive constant.Theorem 23.
LetB be a k-positive holomorphic line bundle on a compact hypercomplex Kähler manifold M. Then(i)
Hp(M,Ωq⊗B)=0, for p+q>2n+k;(ii)
Hp(M,Ωq⊗B)=0, for p>n+[k/2] and any nonnegative integer q.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 M. (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 pointx∈M, which diagonalizes simultaneously the Hermitian form ω(x) and iΘ(x) since both of them are semipositive, such that
(141)ω(x)=i∑j=1n‍μj(x)dzj∧dz¯j,iΘ(x)=i∑j=1n‍νj(x)dzj∧dz¯j.
Without loss of generality, assume that ν1(x)/μ1(x)≤⋯≤ν2n(x)/μ2n(x). Then for any (p,q)-form ξ=∑P,Q‍ξP,QθP∧θ¯Q⊗e, the last big summation vanishes in the formula (111). 〈[ΘJ,ΛJ]ξ,ξ〉 is expressed by the first two big summations in (111) in the following way:
(142)〈[ΘJ,ΛJ]ξ,ξ〉(x)=∑|P|=p,|Q|=q‍(∑p∈Q,1≤p≤2n‍νp(x)μp(x)+∑p+n∈Q,1≤p≤n‍νp(x)μp(x)hhhhhhhhhhhh+∑p∈Q,1≤p≤n‍νp+n(x)μp+n(x)-∑p=12n‍νp(x)μp(x))|ξP,Q|2≥[2(ν1(x)μ1(x)+⋯+νp(x)μp(x))-∑j=12n‍νj(x)μj(x)]|ξ|2.
Observe that if the ratio νj(x)/μj(x) varies small when j varies, for example, in the extreme case when all νj(x)/μj(x) are equal; then [iΘh(E),Λ](x) is positive when p>n. This observation tells us that if we choose the Kähler metric ω properly such that the eigenvalues of iΘh(E) vary mildly relative to ω, then we can deduce the positivity of [iΘh(E),Λ].
SinceB is k-positive if and only if its curvature iΘ is semi-positive with rank iΘ≥n-k, we may take a special choice of the Kähler metric of M with ω~:=ω~I:=iΘ+κω for some positive number κ. Then M is a compact hypercomplex manifold and (M,I) is still Kähler manifold with the new Kähler form ω~. From now on we consider M as a new hypercomplex Kähler manifold with Kähler metric ω~. Correspondingly, we have three new nondegenerate 2-froms, ω~I, ω~J, ω~K. Let Λ~I, Λ~J, Λ~K be the associated adjoint operators of multiplication by ω~I,ω~J,ω~K. We could get the Bochner-Kodaira-Nakano identities with respect to ω~I, ω~J, ω~K as in Section 4. Then the eigenvalues of Θ relative to ω~ are {rj(x)}1≤j≤n with
(143)rj(x)=νj(x)κμj(x)+νj(x)=νj(x)/μj(x)κ+νj(x)/μj(x)=(νj(x)/μj(x))/κ1+((νj(x)/μj(x))/κ)=1-11+((νj(x)/μj(x))/κ).
Fix a point x∈M and assume that rank⁡(Θ|x)=2n-s≥2n-k. Then 0=rs(x)<rs+1(x)≤⋯≤r2n(x). Thus rj(x)=0 for j≤s. If we choose κ→0+, then rj(x)→1 for all s+1≤j≤2n. If p>n+[k/2], then p≥n+(k/2)≥k≥s and
(144)lim⁡κ→0+[2(r1(x)+⋯+rp(x))-(r1(x)+⋯+r2n(x)))=2(p-s)-(2n-s)≥2(p-(n+k2))≥2(1+[k2]-k2)≥1.
Since M is compact, we can use a finite cover by open sets, such that, on each open set, if κ is a sufficiently small positive number, we may have on each open neighborhood and hence everywhere on M the following:
(145)〈[ΘJ,Λ~J]ξ,ξ〉≥12|ξ|2.
Therefore if p>n+[k/2], then [ΘJ,Λ~J] is positive on Ωq,p(B) and (ii) of Theorem 23 follows.Corollary 24.
LetB be a holomorphic k-positive line bundle on a compact hyper-kähler manifold M. Then(i)
Hp(M,Ωq⊗B)=0, for p+q>2n+k;(ii)
Hp(M,Ωq⊗B)=0, for p>n+[k/2] and any nonnegative integer q.In particular, ifB is a positive holomorphic line bundle, we have(i)′
Hp(M,Ωq⊗B)=0, for p+q>2n;(ii)′
Hp(M,Ωq⊗B)=0, for p>n and any nonnegative integer q.In [13], we proved that, on a compact Kähler manifold, any k-ample line bundle is k-positive. So Corollary 24 is also applicable to k-ample line bundle. In particular, if k=0, we get Theorem 2 for algebraic hyper-kähler manifolds.



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*Source: 243236-2014-02-16.xml*