Vanishing Theorems on Compact Hyperk\"ahler Manifolds

We prove that if $B$ is a $k$-positive holomorphic line bundle on a compact hyperk\"ahler manifold $M,$ then $H^p (M,\Omega^q\otimes B)=0$ for $p>n+[\frac{k}{2}]$ and any nonnegative integer $q.$ In a special case $k=0$ and $q=0$ we recover a vanishing theorem of Verbitsky's with a little stronger assumption.

Proof. Let g(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 ω J (X, Y ) = −g(X, JY ) = −ω(X, KY ). Clearly ω J is nondegenerate. It is well-known that dω J = 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 2.3. Let M 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 ∈ T M. Then g is a Lorentz Hermitian metric and for any point x ∈ M there exists a local holomorphic coordinate (w 1 , · · · , w n ) around x such that g = jk (±δ jk + O(|w| 2 ))dw j dw k .
Proof. If moreover I is ω-partible then M is a Käler manifold and there is a proof in [GH] 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 (z j ) at x such that g jk (x) = ±δ jk , in other words we could write locally g = jkl (±δ jk + a jkl z l + a jklz l + O(|z| 2 ))dz j dz k ; ω = i jkl (±δ jk + a jkl z l + a jklz l + O(|z| 2 ))dz j ∧ dz k .
Let us make a holomorphic change of coordinates Then in the new coordinate we have g = jk (±δ jk + l (a jkl w l + a jklwl + b klj w l + b jlkwl + O(|w| 2 )))dw j dw k .
A complex manifold M 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 papers [Ve2], [Ve1], [AV]). 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 [Ob]: Such a connection is called an Obata connection. If M is a hyperkähler manifold, clearly the Levi-Civita connection is exactly the Obata connection of the underlying hypercomplex manifold.
The following proposition is cited from [AV].
Proposition 2.4. Let (M, I, J, K) be a hypercomplex manifold. For any point x ∈ M, there exists a holomorphic (with respect to I) local coordinate (z j ) around x with z j (x) = 0 such that where I 0 , J 0 , K 0 are the constant complex structures.
Let {ξ 1 , Iξ 1 , Jξ 1 , Kξ 1 , · · · , ξ n , Iξ n , Jξ n , Kξ n } 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 ω I locally as Accordingly, . Choose holomorphic coframes relative to the complex structure I, with antiholomorphic coframes Then we have the following pointwise action at x ∈ M : Using holomorphic and antiholomorphic coframes, we could rewrite the 2-forms ω I , ω J , ω K locally as is a holomorphic (2, 0)-form relative the complex structure I.
For convenience, when talking about holomorphic structure of M 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 positions.
Theorem 3.1. There exists a local holomorphic coordinate (z 1 , · · · , z n ) around x such that (3.5) Proof. From [NN], we know there exists a local holomorphic coordinates (z j ) with respect to the complex structure I such that its action is local constant: Idz j = idz j with θ j = dz j , which coincides with the pointwise action we considered in (3.2). By Proposition 2.3, there exists a local holomorphic coordinates (z j ) such that (3.5) holds. By Proposition 2.4 and (3.3), (3.4), (3.10) (3.11) From (3.8) to (3.11) and the definition of ω J and ω K , we conclude (3.6) and (3.7).
Let d : Ω p (M) → Ω p+1 (M) be the de Rham differential operator on M, and let d c = I −1 dI. Note that the complex structures I, J, K on the tangent bundle naturally induces 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 are defined by (3.12) The Dolbeault operators ∂,∂ and d, d c are related by (3.13) Clearly the Dolbeault operators ∂,∂ are determined completely by d and the complex structure I : (3.14) Accordingly, the complex structure J and K also induce complex differential operators where ∂ J = J −1 ∂J and∂ J , ∂ K ,∂ K are similar notions. Like d, ∂,∂, it is easy to check that, the operators d J , d K , ∂ J , ∂ K also satisfy the graded Leibniz rule: for any ξ, η ∈ Ω • (M), where |ξ| is the degree of ξ.
For each ordered set of indices A = {α 1 , · · · , α p }, denote the index length |A| = p; we write θ A = θ α 1 ∧ · · · ∧ θ αp , ,θ A =θ α 1 ∧ · · · ∧θ αp , and denote byÂ = {α p+1 , · · · , α n } the complementary of A so that where τ (A) takes value 1 if A is an even permutation and −1 otherwise. The Hodge star operator * is given by where f is a function and the signature factor Given two (p, q)-forms their pointwise inner product is defined by Since M is compact, we can consider the Hermitian inner product on each Ω p,q (M) defined by For each k = 1, · · · , 2n, let e k : Ω p,q (M) → Ω p+1,q (M) be the wedge operator defined by (3.20) Let i k andī k be the adjoints of e k andē k with respect to the inner product (3.18), respectively. They are called contraction operators. Then for any k, l = 1, · · · 2n, e kīl +ī l e k = 0; (3.21) The three equation above reflect how to commute the actions of wedges and contractions. The following Proposition 3.2 gives the commutation relations between contract actions and complex structure actions on differential forms. Base on them, it is easy to get the commutation relations between the actions of wedges and complex structures Proposition 3.2. As operators on acting on the differential forms, the contractions i k and the complex structures I, J, K satisfying the commuting relations By definition equation (3.15) and the commutation relations in Proposition 3.2, we have the following expressions of differential operators via contraction and wedge operators: (3.27) to Ω * (E) defined by the wedge with the 2-forms ω = ω I , ω J , ω K respectively and Λ = Λ I , Λ J , Λ K their adjoint operators. (3.32) The following identities in Lemma 3.3 called Hodge identities [GH], they play fundamental roles in Kähler geometry. Their proof are reduced from an arbitrary Kähler manifold to the Euclidean Kähler plane via using Proposition 2.3. 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 2.3, a Kähler metric is oscalate order 2 to the Euclidean metric everywhere.
For a proof of this lemma, please refer to [GH], pp111-114.
Proposition 3.4. Let (M, I, J, K) be a compact hypercomplex manifold such that (M, I) is a Kähler manifold, then Proof. The idea of the proof is the same with that of Lemma 3.3, since by Theorem 3.1, the 2-forms ω J and ω K 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 3.3 as in [GH], pp111-114. 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 (3.34). The rest equations are proved in the same way.
By (3.30), we have (3.46) From (3.42),(3.45) and (3.46) we get Note the right hand side of (3.47) is the conjugation of the second equation of (3.28), we arrive at the first equation of (3.34).

Twisted Bochner-Kodaira-Nakano type identities
In this section we will extend the differential operators studied in Sect. 3.1 to act on the 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 are its components. Let {s α } be a holomorphic frame of E. For any E-valued differential forms ξ = α ξ α ⊗s α and η = β η α ⊗ s α of Ω p,q (E), we define their local inner product ξ, η = α,β h αβ ξ α ∧ * η β and global inner product (3.48) are their adjoint operators respectively.
Proof. By Proposition 3.5, The rest equations are proved in the same way.
Let Θφ be the curvature component corresponding to the operator D ′′ D ′φ + D ′φ D ′′ . Let △ ′φ = D ′φ δ ′φ + δ ′φ D ′φ be another twisted Laplacian operator. By using Proposition 3.7, it is easy to prove the following Bochner-Kodaira-Nakano type identity where ϑ ∈ Ω 1,0 (End(E)) is the connection matrix. By the compatible condition of the connection D and the metric h, we have dh = hϑ +θ t h, by comparing the type we get ∂h = hϑ,∂h =θ t h, it follows that and the Chern curvature Θ is given by Let {z j } be the local holomorphic coordinate of M such that the holomorphic coframes in (3.1) are represented by θ j = dz j . Let {s α } be a holomorphic frame, {s α } the dual frame of E. Let (g jk ) and (h αβ ) be the hermitian metrics on M and on E respectively, and their inverses denoted respectively by (g ij ) and (h αβ ). Then the connection and curvature could be expressed by Proof. For any ξ = α ξ α ⊗ s α ∈ Ω p,q (E), From (4.5),(4.6) and (4.9) we conclude the first equation of (4.4). From (3.3) and (3.4), clearly Θ K =∂ϑ K = iΘ J . Since by definition we have Θφ = Θ J .

From (3.3),(3.4) and (4.2) we have the following local expressions of connection and curvature
Using (4.3), we could write the curvature components of Θ J simply as (4.12) Therefore, The proof of the following lemma is simply via using the commuting relation (3.21) (3.23) and (3.22), we omit it here for brevity.
Taking inner products of both sides of the above equation with ξ, and using (4.17), we get immediately the equation (4.18).

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 = ω I . 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 4.6 as an example. If ξ is an arbitrary E-valued (p, q)-form, then an integration by part of the formula If ξ ∈ H p,q (E), then ξ is ∆ ′′ -harmonic and hence D ′′ ξ = D ′′ * ξ = 0 by the Hodge theory.
Thus we get a vanishing cohomology group. Therefore, using (5.1) to prove a vanishing theorem for E-valued Dolbeault cohomology groups, the key point is to find conditions under which the operator [Θ J , Λ J ] is positive definite.
We can see from above reasoning, the second and third formulae of Proposition 4.6 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 positions, 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 (3.59) and one formula of Proposition 4.6 produce different vanishing theorems. Note however, the computations of the proof (though we don't give its proof) of the last formula of Proposition 4.6 is simpler than the other two equations.
By (4.1), the Chern curvature form of E is given by The first Chern class c 1 (E) ∈ H 2 (M, R) is a cohomology class which has a representation via using the Chern curvature form Conversely, any 2-form representing the first Chern class c 1 (E) is in fact the Chern curvature form of some hermitian metric on det E (up to a constant). In local coordinate we have In particular, if E is a line bundle then its curvature form represents its first Chern class up to a constant 1 2π . In [Ya], we introduce the following notion for semipositive holomorphic vector bundles. Base on it and the formula (3.59) for Kähler manifolds, we get some new vanishing theorems.
Definition 5.1. A holomorphic vector bundle E 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 defined on W ⊕s is semipositive and the dimension of its kernel is at most k, where W = T x M (resp. E x ).
Clearly the (0, s)-positivity is equivalent to the Demailly s-positivity [De] and the Nakano positivity [SS] 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 [GH]. A holomorphic vector bundle E of arbitrary rank is called Griffiths k-positive if if it is (k, 1)-positive.
Theorem 5.2. Let M 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) H p (M, E) = 0, for p > k and s ≥ min{2n − p + 1, r}; (ii) If in addition k ≤ 2n − 1, then for s ≥ min{2n − p + 1, r} and any nonnegative integer p, H 2n (M, Ω p ⊗ E) = 0.
A holomorphic line bundle B on M is called k-positive, if there is a hermitian metric on B such that its first Chern class c 1 (B) is semi-positive and has at least n − k positive eigenvalues [Gi], [SS]. If E is a holomorphic line bundle (denoted it by B for the difference), then in Definition 5.1 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 5.3. Let B be a k-positive holomorphic line bundle on a compact hypercomplex Kähler manifold M. Then (i) H p (M, Ω q ⊗ B) = 0, for p + q > 2n + k; (ii) H p (M, Ω q ⊗ B) = 0, for p > n + [ k 2 ] and any nonnegative integer q. Proof. We get (i) by using the Gigante-Girbau's vanishing theorem on Kähler manifolds [Gi], a simple proof is given in Theorem 2.4 of [Ya]. (i) is proved via using (3.59) 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 4.6. Here we give a proof of (ii) in the following paragraph.