Holomorphic Poisson Cohomology

A holomorphic Poisson structure induces a deformation of the complex structure as Hitchin's generalized geometry. Its associated cohomology naturally appears as the limit of a spectral sequence of a double complex. The first sheet of this spectral sequence is the Dolbeault cohomology with coefficients in the exterior algebra of the holomorphic tangent bundle. We identify various necessary conditions on compact complex manifolds on which this spectral sequence degenerates on the level of the second sheet. The manifolds to our concern include all compact complex surfaces, K\"ahler manifolds, and nilmanifolds with abelian complex structures or complex parallelizable manifolds.


Introduction
The algebraic geometry of Poisson brackets over complex manifolds were studied for some time [21]. In recent years, there are significant interests on holomorphic Poisson structures due to their emergence in generalized complex geometry [11] [12] [13]. As such, one could consider their deformations either as complex analytic objects [6] [8] [14], or as generalized complex objects [9]. On the other hand, there are classifications of holomorphic Poisson structures on algebraic surfaces [2], and computations of their related cohomology theory [16].
A holomorphic Poisson structure consists of a holomorphic bi-vector field Λ on a complex manifold such that [[Λ, Λ]] = 0. It is demonstrated in [9] that the deformation of such structure as a generalized complex structure is dictated by the cohomology space ⊕ k H k Λ of the differential operator ∂ Λ = ∂ + [ [Λ, −]]. This observation motivates the authors' desire to find a way to compute the cohomology spaces H k Λ . Since this space is naturally the limit of a spectral sequence of a double complex due to the operators ∂ and [[Λ, −]], it is an obvious question on when and how fast this spectral sequence could degenerate. We will see that the first sheet of this spectral sequence is the Dolbeault cohomology on the manifold with coefficients in the holomorphic polyvectors. As this is a classical object, it becomes desirable to identify the conditions under which the spectral sequence will degenerate at its second level.
In this paper, we identify three situations in which the spectral sequence of the holomorphic Poisson double complex degenerates at its second level. In the next chapter, we will explain the role of holomorphic Poisson structures in generalized geometry, and then set up our computation of cohomology in the context of Lie bi-algebroids and their dual differentials [17]. Our reference for differential calculus of Lie algebroids is the book [18]. We will then set up the holomorphic Poisson double complex and its related spectral sequence.
We will make observation on three very different kinds of complex manifolds: complex parallelizable, nilmanifolds with abelian complex structures, and Kählerian manifolds.
On complex parallelizable manifolds, we have the following result. This result will appear as Theorem 3 in this text. The proof of this result is not hard except for an application of Sakane's work [25] on Dolbeault cohomology on complex parallelizable manifolds because the holomorphic tangent bundle is trivial in this case. A typical example is the complex three-dimensional Iwasawa manifold.
The analysis of nilmanifolds takes significantly more effort. Much are based on our experience with computation of various cohomology on nilmanifolds, starting from Nomizu's work on de Rham cohomology [20], its generalization to Dolbeault cohomology [3] [5] and their generalizations to cohomology with coefficients in holomorphic vector fields [9] [10] [19]. In this paper, we need to deal with cohomology with coefficients in holomorphic poly-vector fields. The result below generalizes what was known for Kodaira surface [22].
Theorem 2. Suppose that M = G/Γ is a 2-step nilmanifold with an abelian complex structure. Suppose that the center of g is real two-dimensional, then for any invariant holomorphic Poisson structure Λ, the Poisson spectral sequence degenerates on the second sheet.
This theorem will appear as Theorem 2. In view of the work in [4] and [24], we expect much room to generalize this theorem. There are also lots of examples.
Finally, to deal with Kählerian manifolds, we adopt an idea from Hitchin in his paper [14] (see also [8]) when he uses a contraction of the holomorphic bivector field Λ to pass a problem at hand to a consideration on Dolbeault cohomology. This contraction yields a homomorphism of double complex. Based on it, we prove the following result.
Theorem 3. Suppose that X is a holomorphic Poisson manifold with complex dimension n. If the Fröhlicher spectral sequence of its complex structure degenerates at the E 1 -level, then its holomorphic Poisson spectral sequence degenerates at the E n -level.
This theorem will appear as Theorem 6. Note that it is well known that the Fröhlicher spectral sequence on a compact complex surface always degenerates [1].
The last theorem could be applied to all compact complex surfaces as well as all compact Kählerian manifolds [27].
determines a generalized complex structure. Here by J * α where α is a 1-form, we mean (J * α)X = α(JX) for any vector field X.
At t = 0, one recovers the generalized complex structure determined by the classical complex structure J on the manifold M. This is an example of deformation of generalized complex structures.
The corresponding bundles as generalized complex structure for J tΛ is the pair of bundles of graphes L tΛ and L tΛ where

Holomorphic Poisson Cohomology
Given a generalized complex structure, the pair of bundles L and L makes a Lie bialgebroid. Via the canonical non-degenerate pairing on the bundle T , the bundle L is complex linearly identified to the dual of L. Therefore, the Lie algebroid differential of L acts on L. It extends to a differential on the exterior algebra of L. For the calculus of Lie bialgebroids, we follow the conventions in [18]. In particular, let ρ be the anchor map on the Lie algebroid L. For any element Γ in C ∞ (M, ∧ k L) and elements a 1 , . . . , a k+1 in C ∞ (M, ∧ k L), the differential of Γ is (∂ L Γ)(a 1 , . . . , a k+1 ) = For example, if L comes from a complex structure, L = T 1,0 M ⊕ T * (0,1) M. The differential is Obviously, when ω is a (0, 1)-form, then ∂ω is the classical ∂-operator in Dolbeault theory. When X is a (1, 0)-vector field, Y a (0, 1)-vector field and ω a (1, 0)-form, Therefore, ∂X is identical to the so-called Riemann-Cauchy operator as in [7]. For future reference, we note the following observation, which could be proved by treating ∂ as a Lie algebroid differential with respect to the Lie algebroid L as we did in the calculus in equation (4) above.

Poisson Double Complex
We have seen that a holomorphic Poisson structure Λ could be treated as a deformation of the complex structure J. While the pair of bundles L Λ and L Λ form a Lie bi-algebroid, so does the pair L and L Λ [17].
Proposition 1 [9] The Lie algebroid differential of L Λ acting on L is given by where by definition ad when v is a smooth section of T 1,0 M and ω is a smooth section of T * (0,1) M.
The operator ∂ Λ extends to act on the exterior algebra of T 1,0 M ⊕ T * (0,1) M: Since Λ is a holomorphic Poisson structure, ∂ Λ • ∂ Λ = 0. Therefore, one has a complex with ∂ Λ being a differential. Starting with k = 0, one has the k-th cohomology of this complex. It is denoted by H k Λ (M). This cohomology space is called the k-th holomorphic Poisson cohomology of Λ; or simply Poisson cohomology.
For the double complex A * , * , ad Λ , ∂ , its associated single complex is K n = ⊕ p+q=n A p,q and the differential is

Lemma 2
The triples A * , * , ad Λ , ∂ form a double complex. Its total cohomology is the holomorphic Poisson cohomology.
The bi-grading yields two spectral sequences abutting to the Poisson cohomology.
We focus on the case when the filtration is given by Equivalently, the zero-th sheet of this spectral sequence E p,q 0 is precisely A p,q . Since ∂ Λ = ∂ + ad Λ and ad Λ does not raise the second index in the bi-degree (p, q), the restriction of the differential on the 0-th sheet is Therefore, the first sheet of this spectral sequence is the Dolbeault cohomology where Θ p is the p-th exterior power of the holomorphic tangent bundle of the complex manifold M.
It follows that the next sheet of this spectral sequence is given by the cohomology of the differential for all p + q ≥ 0, and the second sheet of this spectral sequence is To facilitate further computation, we recall that the differential d p,q 2 is computed as follows. Suppose that 'v' represents a class in H q (M, Θ p ) and ad Λ v represents a zero class in H q (M, Θ p+1 ), it represents an element in E p,q 2 . As ad Λ v represents a zero class, there exists 'w' a section of T p+1,0 M ⊗ T * (0,q−1) M such that ∂w = ad Λ v. Since

Complex Projective Spaces as Examples
Let Λ be any non-trivial holomorphic section of the bundle Θ 2 of the complex pro- Since H q (M, Θ p ) vanishes for all q ≥ 1, the first sheet of the holomorphic Poisson double complex is reduced to its zero-th row: It is apparent that d 1 is not identically equal to zero. It is also apparent that d 2 is identically equal to zero.

Hopf Manifolds
Consider C n with coordinates z = (z 1 , . . . , z n ). Let λ > 1 be a real number. It generates a one-parameter group of automorphism on C n . The quotient of C n \{0} with respect this group is diffeomorphic to the manifold M = S 1 × S 2n−1 . The complex structure on C n descends onto M to define an integrable complex structure.
As a complex manifold, it is a principal elliptic fibration over the complex projective space N = CP n−1 : Denote the fiber by F and the holomorphic vertical vector field by V . We have the following exact sequence of holomorphic vector bundles over the Hopf manifold M: where O X (V ) denote the vertical bundle of the projection π. It is also a bundle trivialized by the vector field V . Explicitly the vector field V is simply the holomorphic vector field generated by the group of dilations.
For all 1 ≤ p ≤ n − 1, the exact sequence (14) generates We It is a global ∂-closed (0, 1)-form on the Hopf manifold M.
• For all 1 ≤ p ≤ n − 1, The proof of this proposition is a straight-forward application of the Leray spectral sequence with respect to the fiberation π. Similar computation could be found in [10] and [15]. Below we outline a proof. First of all, by virtue of the dimension of the fiber and the global (0, 1)-form ω on the manifold M, one proves that the direct image sheaves of R q π * O M = 0 for q ≥ 2, and By the Projective Formula, for all k ≥ 0, R q π * π * Θ k N = 0 for q ≥ 2, and As the second sheet of the Leray spectral sequence for π * Θ k N is Recall that N is the complex projective space CP n−1 . Therefore, the above cohomology spaces vanish except when p = 0. It follows immediately the differential d p,q 2 is identically zero, and hence the Leray spectral sequence degenerates to By the exact sequence (16), one concludes immediately, that for all p ≥ 0 and q ≥ 2, By chasing diagram, one could actually show that the induced long exact sequence of (16) splits. The claim in the proposition above follows.
Given the proposition above, one notes that the non-zero terms of the first sheet of the Poisson double complex with respect to any holomorphic Poisson structure on the Hopf manifold are contained in the zero-th and the first rows only. To compute the differential on the second sheet explicitly, note that the first sheet is given by To prepare our computation, note that when we treat CP n−1 as a homogeneous space, then its space of holomorphic vector fields is sl(n, C). They are generated by linear action of SU(n) on C n .
Let A ∈ sl(n, C), and define on C n the function By homogeneity, this function is well-defined on the Hopf manifold M. It is straight- Consider the bi-vector field Λ = V ∧ A. Since A and V commute, Λ is a holomorphic Poisson structure. We claim that its associated Poisson double complex degenerates at the E 2 -level.
Since the only non-trivial elements in the first sheet of the double complex are in the zero-th and the first row, E p,q 2 vanishes except possibly when (p, q) = (p, 0) and (p, q) = (p, 1). Therefore, the only possibly non-trivial differential on the second sheet are d p, 1 2 . According to the structure of the cohomology spaces, each term in Note that the first summand is equal to zero because L V f A = 0. The second summand is equal to zero because V commutes with every element in H 0 (M, Θ p ).
The third summand is equal to zero by skew-symmetry of exterior multiplication.
The last summand is equal to zero because as noted above when the term [[ad Λ , ω∧Γ]] represents the zero class in In the rest of this article, we investigate the degeneracy of holomorphic Poisson spectral sequence in various special settings.

Nilmanifolds
A compact manifold M is called a nilmanifold if there exists a simply-connected nilpotent Lie group G and a lattice subgroup Γ such that M is diffeomorphic to G/Γ. We denote the Lie algebra of G by g, and the center of g by Z(g). The step of the nilmanifold is the nilpotence of the Lie algebra g. Denote ∧ k g 1,0 and ∧ k g * (0,1) respectively by g k,0 and g * (0,k) .

Cohomology of Nilmanifolds
Now assume that the Lie algebra g is 2-step nilpotent, i.e. [g, g] ⊂ Z(g). In such a case, we call M = G/Γ a 2-step nilmanifold [19].
The concerned cohomology is The inclusion map yields an inclusion of cohomology: Theorem 1 On a 2-step nilmanifold M with an invariant abelian complex structure, the inclusion g p,0 ⊗ g * (0,q) in C ∞ (M, ∧ p T 1,0 M ⊗ ∧ q T * (0,1) M) induces an isomorphism of cohomology. In other words, We need some preparations before we give a proof for this theorem.
Let C be the center of G and ψ : G → G/C the quotient map. Since G is 2-step nilpotent, G/C is abelian. Consider M = G/Γ and N = ψ(G)/ψ(Γ). We have a holomorphic fibration Ψ : M → N whose fiber is isomorphic to F = C/(C ∩ Γ).
Moreover, the co-boundary operator∂ : g 1,0 → g * (0,1) ⊗ g 1,0 satisfies Lemma 3 Let Θ N be the tangent sheaf of N, the quotient manifold. Then Proof: By Lemma 4 in [19], we have R q Ψ * (O M ) = c * (0,q) ⊗ O N . Thus, by the projective formula, Proof: The proof is essentially the same as that of Lemma 5 in [19]. Here we give a sketch. First note that N is an abelian variety. It follows that Θ N is holomorphically trivial, and is isomorphic to t 1,0 . Therefore, H k (N, ∧ l Θ N ) = t * (0,k) ⊗ t l,0 .
Moreover, the associated graded spaces are: Accordingly, we have a filtration of the co-chain complex C • = T * (0,•) ⊗ ∧ l Θ M : . Thus there associates a spectral sequence: E p,q r , which starts with and d 0 =∂. It follows from Lemma 4 that we have The associated d 1 : E p,q 1 → E p+1,q 1 should be exactlȳ Here we use the fact that g is 2-step nilpotent, namely the relations in (25) and (26). Hence 1,0) ) .
The operator d 2 : E p,q 2 → E p+2,q−1 2 is essentially that of∂. So the previous expression implies that d 2 = 0. So E p,q 2 = E p,q ∞ . Therefore, one has as required.

Degeneracy of Holomorphic Poisson Double Complex
As a consequence of Theorem 1, where H q (g p,0 ) is given in (18).
Theorem 2 Suppose that M = G/Γ is a 2-step nilmanifold with an abelian complex structure. Suppose that the center of g is real two-dimensional, then for any invariant holomorphic Poisson structure Λ, the Poisson spectral sequence degenerates on the second sheet.

Examples
We consider examples of holomorphic Poisson structures that satisfies the conditions in Theorem 2. In particular, let W be an element in c 1,0 and T an element in t 1,0 .
The first example of non-abelian nilmanifold is the Kodaira surface. As a nilmanifold, it is covered by the product of the real three-dimensional Heisenberg group H 3 and the one-dimensional trivial additive group R 1 . Computation on this manifold was done extensively in [22]. It is known to have an invariant holomorphic symplectic structure. Using the usual transformation from symplectic structure to Poisson structure, one obtains a holomorphic Poisson structure on such a Kodaira surface such that the Poisson bi-vector field is of the type W ∧ T , and hence satisfies the conditions in Theorem 2. We skip the details for this case, and refer readers to [22].
Six-dimensional 2-step nilmanifolds with abelian complex structures were studied extensively in [5] [19] [26]. The underlying algebraic structure is classified. In particular, let R n denotes the n-dimensional abelian group and H n the n-dimensional real Heisenberg group, then the 2-step nilpotent groups with abelian complex struc- , the real six-dimensional Iwasawa group W 6 and one additional group P 6 which we will describe with a little more details below. Since R 6 is abelian and H 3 × R 3 covers the product of a Kodaira surface and an elliptic curve, they are not new examples in a strict sense. We consider the four remaining cases in their respective series in all admisssible dimensions, namely and P 4n+2 . We demonstrates that they will each provide an example for Theorem 2.
Define a Lie bracket by for all k. Then {X k , Y k , Z 1 : 1 ≤ k ≤ n} span the (2n + 1)-dimensional Heisenberg algebra h 2n+1 . With Z 2 , g is the one-dimensional trivial extension of the Heisenberg algebra. Its center is spanned by Z 1 and Z 2 . Now, define a linear map by It defines an abelian complex structure.

W 4n+2
It is a 4n + 2-dimensional nilpotent group. On Lie algebra level, let be a basis such that the non-zero structure equations are given by We obtain an abelian complex structure if we insist J 2 = −identity, and

P 4n+2
It is again a 4n + 2-dimensional nilpotent group. On Lie algebra level, it is a degeneration of the structure in (35). In particular, with respect to the basis as given in (34), the non-zero structure equations are given by An abelian complex structure is defined by the requirements in (36).

Complex Parallelizable Nilmanifolds
A nilmanifold is complex parallelizable if there is an invariant complex structure J such that for all X, Y in g, In this case, g is the underlying real Lie algebra of a complex Lie algebra, and Dually, when ∂ is the Chevalley-Eilenberg differential on g * (1,0) , then (⊕g * (1,0) , ∂) forms an exterior differential algebra. So is the conjugate. Let H q ∂ (g * (0,1) ) denote the q-th cohomology of the differential algebra (⊕g * (0,q) , ∂). It is proved in [25] that the Dolbeault cohomology H p,q (M) is isomorphic to the H q ∂ (g * (0,1) ) ⊗ g * (p,0) . Since the holomorphic tangent bundle for a complex parallelizable manifold is trivialized by the complex vector fields of the algebra g 1,0 , Suppose that Λ is an invariant holomorphic Poisson structure on M. The above observation implies that the first sheet of the Poisson double complex is given by Since the exterior differential of g * (0,1) is contained in g * (0,2) , ad Λ (ω) = 0 for all ω ∈ g * (0,1) . If j Ω j ⊗ θ j represents an element in H q ∂ (g * (0,1) ) ⊗ g p,0 where Ω j for appropriate number of j forms a basis for H q ∂ (g * (0,1) ), and θ j ∈ g p,0 , then Therefore, it represents a zero class in H q+1 ∂ (g * (0,1) ) ⊗ g p,0 if and only if ad Λ (θ j ) = 0 for each j. It follows that d p,q 2 ≡ 0. Moreover, (g p,0 , ad Λ ) form a complex. Let

Complex Three-dimensional Iwasawa Manifolds
Let G be the complex Heisenberg group of 3 × 3 matrices of the form Let H be the co-compact lattice of matrices whose entries are Gaussian integers.

Lichnerowicz Homomorphism
On any complex manifold, there is a well known double complex, the Dolbeault bi-complex. Recall that d = ∂ + ∂. Define Due to the integrability of the complex structure, we have the double complex . Given a holomorphic Poisson structure Λ, contraction with −Λ defines a map from B 1,0 to A 2,0 . We extend to the map φ : B p,q → A p,q as follows.
By construction, it is apparent that φ maps B p,q to A p,q . The rest of this section is to prove the following observation.

Theorem 4
The map φ is a homomorphism from the Dolbeault double complex to the Poisson double complex.
It suffices to verify the above identities on B 0,0 , B 1,0 and B 0,1 . For any function f , As ∂f is a (0, 1)-form, it is clear that φ • ∂f = ∂f . To work on the other identity, we do it locally, and there exists (1, 0) holomorphic vector fields X j and Y j such that at least locally, Λ = j X j ∧ Y j . Then On the other hand, Therefore, the identities in (41) are satisfied when they are applied to smooth functions on the manifold.
On the other hand, we treat ∂ as the Lie algebroid differential for the bundle L, By (43) and (44), we conclude that the identifies in (41) holds when it is tested against any (1, 0)-forms.
The proof of Theorem 4 is now complete.

Applications
The first identity in (41) implies that the bundle map φ induces a complex linear homomorphism φ : H q (X, Ω p ) −→ H q (X, Θ p ) .
The second identity implies that this homomorphism fits into the following commutative diagram: where Ω p is the holomorphic bundle of (p, 0)-forms. The map is defined so that when p = 0, φ is an identity map.

Lemma 11
Suppose that X is holomorphic Poisson manifold such that the Fröhlicher spectral sequence degenerates at the E 1 -level, then for spectral sequence of the holomorphic Poisson double complex E 0,q n = E 0,q 1 = H q (X, O) , and d 0,q n ≡ 0 for all q and for all n ≥ 2.
Proof: As noted above, when p = 0, the homomorphism φ is an isomorphism. The E 0,q 1 -term for the Fröhlicher spectral sequence is H q (X, O). When d p,q 1 = ∂ is identically equal to zero on it, then d 0,q 1 = ad Λ is identically equal to zero on the E 0,q 1 -term for the Poisson spectral sequence, which is H q (X, O) as well. It follows that E 0,q 2 = H q (X, O) . Suppose that ω is a ∂-closed (0, q)-form representing an element in H q (X, O) .
Since d 1 ω represents the zero element in H q (X, Θ), there exists a section A of A 1,q−1 such that ad Λ (ω) = ∂A.
It is now apparent that by induction, E 0,q n+1 = E 0,q n . After all, we have seen that if ω is in H q (X, O), then under the current assumption, ∂ω + ad Λ (ω) = 0.
Theorem 5 The holomorphic Poisson spectral sequence on any compact holomorphic Poisson surface degenerates at E 2 -level.
Proof: The first sheet of the holomorphic Poisson spectral sequence of a compact complex surface is: