Theorem of existence and completeness for holomorphic Poisson structures
aa r X i v : . [ m a t h . AG ] D ec THEOREM OF EXISTENCE AND COMPLETENESS FOR HOLOMORPHICPOISSON STRUCTURES
CHUNGHOON KIM
Abstract.
In this paper, we define a concept of a family of compact holomorphic Poisson manifoldson the basis of Kodaira-Spencer’s deformation theory and deduce the integrability condition. Weprove an analogue of their ‘Theorem of existence for complex analytic structures’ under someanalytic assumption, and establish an analogue of their ‘Theorem of completeness for complexanalytic structures’ in the context of holomorphic Poisson deformations.
Contents
1. Introduction 12. Families of compact holomorphic Poisson manifolds 33. Infinitesimal deformations 54. Integrability condition 85. Theorem of existence for holomorphic Poisson structures 186. Theorem of completeness for holomorphic Poisson structures 21References 361.
Introduction
In this paper, we study deformations of holomorphic Poisson structures in the framework of Ko-daira and Spencer’s deformation theory of complex analytic structures([KS58a],[KS60]). The maindifference from Kodaira and Spencer’s deformation theory is that for deformations of a compactholomorphic Poisson manifold, we deform not only its complex structures, but also holomorphicPoisson structures. We will briefly review Kodaira-Spencer’s main idea and show how we canextend their idea in the context of deformations of holomorphic Poisson structures.Kodaira and Spencer’s main idea of deformations of complex analytic structures is as follows[Kod05, p.182]. A n -dimensional compact complex manifold M is obtained by glueing domains U , ..., U n in C n : M = S nj =1 U j where U = { U j | j = 1 , ..., n } is a locally finite open covering of M ,and that each U j is a polydisk: U j = { z j ∈ C n || z j | < , ..., | z nj | < } and for p ∈ U j ∩ U k , the coordinate transformation f jk : z k → z j = ( z j , ..., z nj ) = f jk ( z k )transforming the local coordinates z k = ( z k , ..., z nk ) = z k ( p ) into the local coordinates z j =( z j , ..., z nj ) = z j ( p ) is biholomorphic. According to Kodaira, “A deformation of M is considered to be the glueing of the same polydisks U j via dif-ferent identification. In other words, replacing f αjk ( z k ) by the functions f αjk ( z k , t ) = f αjk ( z k , t , ..., t m ) , f jk ( z k ,
0) = f αjk ( z k ) of z k , and the parameter t = ( t , ..., t m ) , weobtain deformations M t of M = M by glueing the polydisks U , ..., U n by identifying z k ∈ U k with z j = f jk ( z k , t ) ∈ U j ” This paper is based on the first part of the author’s Ph.D. thesis [Kim14]. While writing this paper, the authorwas partially supported by NRF grant 2011-0027969.
We extend the main idea of Kodaira-Spencer in the context of deformations of holomorphic Pois-son structures. A n -dimensional compact holomorphic Poisson manifold M is a compact complexmanifold such that the structure sheaf O M is a sheaf of Poisson algebras (we refer to [LGPV13]for general information on Poisson geometry). The holomorphic Poisson structure is encoded in aholomorphic section (a holomorphic bivector field) Λ ∈ H ( M, ∧ Θ M ) with [Λ , Λ] = 0, where Θ M isthe sheaf of germs of holomorphic vector fields on M and the bracket [ − , − ] is the Schouten bracketon M . In the sequel a holomorphic Poisson manifold will be denoted by ( M, Λ). For deformationsof a compact holomorphic Poisson manifold ( M, Λ), we extend the idea of Kodaira and Spencer. A n -dimensional compact holomorphic Poisson manifold is obtained by glueing the domains U , ..., U n in C n : M = S nj =1 U j where U = { U j | j = 1 , ..., n } is a locally finite open covering of M and each U j is a polydisk U j = { z j ∈ C n || z j | < , ..., | z nj | < } equipped with a holomorphic bivector field Λ j = P nα,β =1 g jαβ ( z j ) ∂∂z αj ∧ ∂∂z βj such that g jαβ ( z j ) = − g jβα ( z j ) with [Λ j , Λ j ] = 0 on U j and for p ∈ U j ∩ U k , the coordinate transformation f jk : z k → z j = ( z j , ..., z nj ) = f jk ( z k )transforming the local coordinates z k = ( z k , ..., z nk ) = z k ( p ) into the local coordinates z j =( z j , ..., z nj ) = z j ( p ) is a biholomorphic ‘Poisson’ map.Deformations of a compact holomorphic Poisson manifold ( M, Λ) is the glueing of the Poissonpolydisks ( U j , Λ j ( t )) parametrized by t . That is, replacing f αjk ( z k ) by f αjk ( z k , t )( f jk ( z k ,
0) = f αjk ( z k )of z k ), replacing Λ j = P nα,β =1 g jαβ ( z j ) ∂∂z αj ∧ ∂∂z βj by Λ j ( t ) = P nα,β =1 g jαβ ( z j , t ) ∂∂z αj ∧ ∂∂z βj with[Λ j ( t ) , Λ j ( t )] = 0 and Λ j (0) = Λ j , and the parmeter t = ( t , ..., t m ), we obtain deformations( M t , Λ t ) by gluing the Poisson polydisks ( U , Λ ( t )) , ..., ( U n , Λ n ( t )) by identifying z k ∈ U k with z j = f jk ( z k , t ) ∈ U j . The work on deformations of holomorphic Poisson structures is based on thisfundamental idea.In section 2, we define a family of compact holomorphic Poisson manifolds, called a Poissonanalytic family in the framework of Kodaira-Spencer’s deformation theory. In other words, whenwe ignore Poisson structures, a family of compact holomorphic Poisson manifolds is just a familyof compact complex manifolds in the sense of Kodaira and Spencer. So deformations of com-pact holomorphic Poisson manifolds means that we deform complex structures as well as Poissonstructures.In section 3, we show that infinitesimal deformations of a holomorphic Poisson manifold ( M, Λ )in a Poisson analytic family are encoded in the first ‘degree-shifted by 1’ truncated holomorphicPoisson cohomology group. More precisely, an infinitesimal deformation is realized as an elementin the first hypercohomology group H ( M, Θ • M ) of a complex of sheaves Θ • M : Θ M → ∧ Θ M →· · · → ∧ n Θ M → , − ]. Analogously to deformations of complex structure, we defineso called Poisson Kodaira-Spencer map where the Kodaira-Spencer map is realized as a componentof the Poisson Kodaira-Spencer map.In section 4, we study the integrability condition for a Poisson analytic family. Kodaira showedthat given a family of deformations of a compact complex manifold M , locally the family is rep-resented by a C ∞ vector (0 , ϕ ( t ) ∈ A , ( M, T M ) with ϕ (0) = 0 satisfying the integrabilitycondition ¯ ∂ϕ ( t ) − [ ϕ ( t ) , ϕ ( t )] = 0 (see [Kod05] § T M is the holomorphic tangent bundleof M and we use the notation A , ( M, T M ) instead of L , ( T M ) in [Kod05]). We show that givena family of deformations of a compact holomorphic Poisson manifold ( M, Λ ), locally the family isrepresented by a C ∞ vector (0 , ϕ ( t ) with ϕ (0) = 0 and a C ∞ bivector Λ( t ) ∈ A , ( M, ∧ T M )with Λ(0) = Λ satisfying the integrability condition [Λ( t ) , Λ( t )] = 0 , ¯ ∂ Λ( t ) − [Λ( t ) , ϕ ( t )] = 0, and¯ ∂ϕ ( t ) − [ ϕ ( t ) , ϕ ( t )] = 0. Replacing ϕ ( t ) by − ϕ ( t ) and putting Λ ′ ( t ) := Λ( t ) − Λ so that we haveΛ ′ (0) = 0, the integrability condition is equivalent to L ( ϕ ( t )+Λ ′ ( t ))+ [ ϕ ( t )+Λ ′ ( t ) , ϕ ( t )+Λ ′ ( t )] = 0where L = ¯ ∂ + [Λ , − ]. Then ϕ ( t ) + Λ ′ ( t ) is a solution of Maurer Cartan equation of the followingdifferential graded Lie algebra g = ( M i ≥ g i , g i = M p + q − i,p ≥ ,q ≥ A ,p ( M, ∧ q T M ) , L = ¯ ∂ + [Λ , − ] , [ − , − ]) , (1.0.1) HEOREM OF EXISTENCE AND COMPLETENESS FOR HOLOMORPHIC POISSON STRUCTURES 3 where [ − , − ] is the Schouten bracket on M , and A ,p ( M, ∧ q T M ) is the global section of A ,p ( ∧ q T M )the sheaf of germs of C ∞ -section of ∧ p ¯ T ∗ M ⊗ ∧ q T M . Here ¯ T ∗ M is the dual bundle of antiholomor-phic tangent bundle ¯ T M (see [Kod05] p.108). We remark that the integrability condition wasproved in more general context in the language of generalized complex geometry (See [Gua11]).As H ( M, Θ M ) is realized as a subspace of the second cohomology group of a compact complexmanifold M in the sense of generalized complex geometry, H ( M, Θ • M ) is realized as a subspace ofthe second cohomology group of a compact holomorphic Poisson manifold ( M, Λ ) in the sense ofgeneralized complex geometry. In this paper, we deduce the integrability condition by extendingKodaira-Spencer’s original approach, that is, by starting from a concept of a geometric family (aPoisson analytic family).In section 5, under some analytic assumption, we establish an analogous theorem to the followingtheorem of Kodaira and Spencer ([KNS58],[Kod05] p.270). Theorem 1.0.2 (Theorem of existence for complex analytic structures) . Let M be a compact com-plex manifold and suppose H ( M, Θ) = 0 . Then there exists a complex analytic family ( M , B, ω ) with ∈ B ⊂ C m satisfying the following conditions: (1) ω − (0) = M (2) The Kodaira-Spencer map ρ : ∂∂t → (cid:0) ∂M t ∂t (cid:1) t =0 with M t = ω − ( t ) is an isomorphism of T ( B ) onto H ( M, Θ M ) : T ( B ) ρ −→ H ( M, Θ M ) . Similarly, we prove ‘Theorem of existence for deformations of holomorphic Poisson structures’(see Theorem 5.1.1).
Theorem 1.0.3 (Theorem of existence for holomorphic Poisson structures) . Let ( M, Λ ) be acompact holomorphic Poisson manifold such that the associated Laplacian operator (cid:3) ( induced fromthe operator ¯ ∂ + [Λ , − ]) is strongly elliptic and of diagonal type. Suppose that H ( M, Θ • ) = 0 .Then there exists a Poisson analytic family ( M , Λ , B, ω ) with ∈ B ⊂ C m satisfying the followingconditions: (1) ω − (0) = ( M, Λ )(2) The Poisson Kodaira-Spencer map ϕ : ∂∂t → (cid:16) ∂ ( M t , Λ t ) ∂t (cid:17) t =0 with ( M t , Λ t ) = ω − ( t ) is anisomorphism of T ( B ) onto H ( M, Θ • M ) : T B ρ −→ H ( M, Θ • M ) . The proof is rather formal. The proof follows from the Kuranishi’s method presented in [MK06].The reason for the assumption on the associated Laplacian operator (cid:3) (induced from the operator¯ ∂ + [Λ , − ]) is for applying the Kuranishi’s method in the holomorphic Poisson context.In section 6, we establish an analogous theorem to the following theorem of Kodaira and Spencer([KS58b],[Kod05] p.284). Theorem 1.0.4 (Theorem of completeness for complex analytic structures) . Let ( M , B, ω ) be acomplex analytic family of deformations of a compact complex manifold M = ω − (0) , B a domainof C m containing . If the Kodaira-Spencer map ρ : T ( B ) → H ( M , Θ M ) is surjective, thecomplex analytic family ( M , B, ω ) is complete at ∈ B . Similarly, we prove the following theorem which is an analogue of ‘Theorem of completeness’ byKodaira-Spencer.
Theorem 1.0.5 (Theorem of completeness for holomorphic Poisson structures) . Let ( M , Λ M , B, ω ) be a Poisson analytic family of deformations of a compact holomorphic Poisson manifold ( M, Λ ) = ω − (0) , B a domain of C m containing . If the Poisson Kodaira-Spencer map ϕ : T ( B ) → H ( M, Θ • M ) is surjective, the Poisson analytic family ( M , Λ M , B, ω ) is complete at ∈ B . Families of compact holomorphic Poisson manifolds
Definition 2.0.6. ( compare [Kod05] p.59 ) Suppose that given a domain B ⊂ C m , there is a set { ( M t , Λ t ) | t ∈ B } of n -dimensional compact holomorphic Poisson manifolds ( M t , Λ t ) , depending on t = ( t , ..., t m ) ∈ B . We say that { ( M t , Λ t ) | t ∈ B } is a family of compact holomorphic Poissonmanifolds or a Poisson analytic family of compact holomorphic Poisson manifolds if there exists aholomorphic Poisson manifold ( M , Λ) and a holomorphic map ω : M → B satisfing the followingproperties CHUNGHOON KIM (1) ω − ( t ) is a compact holomorphic Poisson submanifold of ( M , Λ) for each t ∈ B . (2) ( M t , Λ t ) = ω − ( t )( M t has the induced Poisson holomorphic structure Λ t from Λ) . (3) The rank of Jacobian of ω is equal to m at every point of M .We will denote a Poisson analytic family by ( M , Λ , B, ω ) . We also call ( M , Λ , B, ω ) a Poissonanalytic family of deformations of a compact holomorphic Poisson manifold ( M t , Λ t ) for eachfixed t ∈ B . Remark 2.0.7.
When we ignore Poisson structures, a Poisson analytic family ( M , Λ , B, ω ) is acomplex analytic family ( M , B, ω ) in the sense of Kodaira-Spencer ( see [Kod05] p.59 ) . Remark 2.0.8.
Given a Poisson analytic family ( M , Λ , B, ω ) as in Definition . . , we can choosea locally finite open covering U = {U j } of M such that U j are coordinate polydisks with a systemof local complex coordinates { z , ..., z j , ... } , where a local coordinate function z j : p → z j ( p ) on U j satisfies z j ( p ) = ( z j ( p ) , ..., z nj ( p ) , t , ..., t m ) , and t = ( t , ..., t m ) = ω ( p ) . Then for a fixed t ∈ B , { p ( z j ( p ) , ..., z nj ( p )) |U j ∩ M t = ∅} gives a system of local complex coordinates on M t . In termsof these coordinates, ω is the projection given by ( z j , t ) = ( z j , ..., z nj , t , ..., t m ) → ( t , ..., t m ) . For j, k with U j ∩U k = ∅ , we denote the coordinate transformations from z k to z j by f jk : ( z k , ..., z nk , t ) → ( z j , ..., z nj , t ) = f jk ( z k , ..., z nk , t )( for the detail, see [Kod05] p.60 ) .On the other hand, since ( M t , Λ t ) ֒ → ( M , Λ) is a holomorphic Poisson submanifold for each t ∈ B and M = S t M t , the holomorphic Poisson structure Λ on M can be expressed in terms oflocal coordinates as Λ = P nα,β =1 g jαβ ( z j , ..., z nj , t ) ∂∂z αj ∧ ∂∂z βj on U j , where g jαβ ( z j , t ) = g jαβ ( z j , ..., z nn , t ) is holomorphic with respect to ( z j , t ) with g jαβ ( z j , t ) = − g jβα ( z j , t ) . For a fixed t , the holomorphicPoisson structure Λ t on M t is given by P nα,β =1 g jαβ ( z j , ..., z nj , t ) ∂∂z αj ∧ ∂∂z βj on U j ∩ M t . Remark 2.0.9.
Let ( M , Λ , B, ω ) be a Poisson analytic family. Let ∆ be an open set of B . Thenthe restriction ( M ∆ = ω − (∆) , Λ | M ∆ , ∆ , ω | M ∆ ) is also a Poisson analytic family. We will denotethe family by ( M ∆ , Λ ∆ , ∆ , ω ) . Example 2.0.10 (complex tori) . ([KS58a] p. Let S be the space of n × n matrices s = ( s αβ ) with det( Im ( s )) > , where α denotes the row index and β the column index, and Im ( s ) is theimaginary part of s . For each matrix s ∈ S we define an n × n matrix ω ( s ) = ( ω αj ( s )) by ω αj ( s ) = ( δ αj , for ≤ j ≤ ns αβ , for j = n + β, ≤ β ≤ n Let G be the discontinuous abelian group of analytic automorphisms of C n × S generated by g j : ( z, s ) → ( z + ω j ( s ) , s ) , j = 1 , ..., n, where ω j ( s ) = ( ω j ( s ) , ..., ω αj ( s ) , ..., ω nj ( s )) is th j -thcolumn vector of ω ( s ) . The quotient space M = C n × S/G and π : M → S induced from thecanonical projection C n × S → S forms a complex analytic family of complex tori. We will put aholomorphic Poisson structure on M to make a Poisson analytic family. A holomorphic bivectorfield of the form Λ = P ni,j =1 f ij ( s ) ∂∂z i ∧ ∂∂z j on C n × S where f ij ( s ) = f ij ( z, s ) are holomorphicfunctions on C n × S , independent of z , is a G -invariant bivector field on C n × S . So this inducesa holomorphic bivector field on M . Since f ij ( s ) are independent of z , we have [Λ , Λ] = 0 . So ( M , Λ , S, π ) is a Poisson analytic family. Example 2.0.11 (Hirzebruch-Nagata surface) . ([SU02] p. Take two C × P C × C and write thecoordinates as ( u, ( ξ : ξ ) , t ) , ( v, ( η : η ) , t )) , respectively, where u, v, t are the coordinates of C and ( ξ : ξ ) , ( η : η ) are the homogeneous coordinates of P C . By patching two C × P C × C together byrelation u = 1 /v, ( ξ : ξ ) = ( η : v m η + tv k η ) , m − ≤ k ≤ m, where m, k are natural numbers t = t, we obtain a complex analytic family π : S → C which is induced from the natural projection C × P C × C → C to the third component. We will put a holomorphic Poisson structure Λ on S sothat ( S , Λ , C , π ) is a Poisson analytic family. S has four affine covers. For one C × P C × C with HEOREM OF EXISTENCE AND COMPLETENESS FOR HOLOMORPHIC POISSON STRUCTURES 5 coordinate ( u, ( ξ : ξ ) , t ) , we have two affine covers, namely, C × C × C and C × C × C . They are gluedvia C × ( C − { } ) × C and C × ( C − { } ) × C by ( u, x = ξ ξ , t ) ( u, y = ξ ξ , t ) = ( u, x , t ) . Similarlyfor another C × P C × C , two affine covers are glued via C × ( C − { } ) × C and C × ( C − { } ) × C by ( v, w = η η , t ) ( v, z, t ) = ( v, w = η η , t ) . We put holomorphic Poisson structures on each fouraffine covers which define a global bivector field Λ with [Λ , Λ] = 0 on S . On ( u, x, t ) coordinate,we give g ( t ) x ∂∂u ∧ ∂∂x , where g ( t ) is any holomorphic function depending only on t . On ( u, y, t ) coordinate, we give − g ( t ) ∂∂u ∧ ∂∂y . On ( v, w, t ) coordinate, we give − g ( t ) v k − m +2 ( wv m − k + t ) ∂∂v ∧ ∂∂w .On ( v, z, t ) coordinate, we give g ( t ) v k − m +2 ( v m − k + tz ) ∂∂v ∧ ∂∂z . Then ( S , Λ , C , π ) is a Poissonanalytic family. Example 2.0.12 (Hopf surfaces) . We construct an one parameter Poisson analytic family ofgeneral Hopf surfaces. An automorphism of W × C given by g : ( z , z , t ) → ( az + tz m , bz , t ) where < | a | ≤ | b | < and b m − a = 0 ( i.e a = b m ) , generates an infinite cyclic group G ,which properly discontinuous and fixed point free. Hence M := W × C /G is a complex manifold.Since the projection of W × C to C commutes with g , it induces a holomorphic map ω of M to C . So ( M , C , ω ) is a complex analytic family. Since g n is given by g n : ( z , z , t ) → ( z ′ , z ′ z , t ′ ) =( a n z + na n − tz m , b n z , t ) , we have ∂∂z = a n ∂∂z ′ , ∂∂z = mna n − tz m − ∂∂z ′ + b n ∂∂z ′ , ∂∂z ∧ ∂∂z = a n b n ∂∂z ′ ∧ ∂∂z ′ Then f ( t ) z m +12 ∂∂z ∧ ∂∂z where f ( t ) is any holomorphic function, independent of z , is a G -invariantholomorphic bivector field on W × C and so define a holomorphic Poisson structure on M . Hence ( M , f ( t ) z m +12 ∂∂z ∧ ∂∂z , C , ω ) is a Poisson analytic family of Poisson Hopf surfaces. Infinitesimal deformations
Infinitesimal deformations and truncated holomorphic Poisson cohomology.
In this subsection, we show that given a Poisson analytic family ( M , Λ , B, ω ), an infinitesimaldeformation of a compact holomorphic Poisson manifold ω − ( t ) = ( M t , Λ t ) with dimension n iscaptured by an element in the first hypercohomology group of the complex of sheaves Θ • M t : Θ M t →∧ Θ M t → · · · → ∧ n Θ M t → t , − ] analogously to how an infinitesimal deformation of acompact complex manifold M t is captured by an element in the first cohomology group H ( M t , Θ t ).Let ( M, Λ ) be a compact holomorphic Poisson manifold and consider the complex of sheavesΘ • M : Θ M [Λ , − ] −−−−→ ∧ Θ M [Λ , − ] −−−−→ · · · [Λ , − ] −−−−→ ∧ n Θ M → M is the sheaf of germs of holomorphic vector fields on M . Let U = { U j } be sufficientlyfine open covering of M such that U j are coordinate polydisks of M , that is, U j = { ( z j , ..., z nj ) ∈ C n || z αj | < r αj , α = 1 , ..., n } where z j = ( z j , ...z nj ) is a local coordinate on U j and r αj > . .
1) bythe following ˘Cech resolution (see [EV92] Appendix). Here δ is the ˘Cech map. [Λ , − ] x C ( U , ∧ Θ M ) − δ −−−−→ · · · [Λ , − ] x [Λ , − ] x C ( U , ∧ Θ M ) δ −−−−→ C ( U , ∧ Θ M ) − δ −−−−→ · · · [Λ , − ] x [Λ , − ] x [Λ , − ] x C ( U , Θ M ) − δ −−−−→ C ( U , Θ M ) δ −−−−→ C ( U , Θ M ) − δ −−−−→ · · · Definition 3.1.2.
We say that the i -th ‘degree-shifted by ’ truncated holomorphic Poisson co-homology group of a holomorphic Poisson manifold ( M, Λ ) is the i -th hypercohomology groupassociated with the complex of sheaves (3 . . , and is denoted by H i ( M, Θ • M ) . CHUNGHOON KIM
Remark 3.1.3. In [ELW99] , the holomorphic Poisson cohomology for a holomorphic Poisson man-ifold ( M, Λ ) is defined by the i -th hypercohomology group of complex of sheaves O M → Θ M →∧ Θ M → · · · → ∧ n Θ M → induced by [Λ , − ] . Since there is no role of the structure sheaf O M in deformations of compact holomorphic Poisson manifolds, we truncate the complex of sheaves toget → Θ M → ∧ Θ M → · · · ∧ n Θ M → . In [Kim14] , the author used the expression HP i ( M, Λ ) for the i -th truncated holomorphic Poisson cohomology group to maintain notational consistencywith [Nam08] by which this present work was inspired. However we shift the degree after truncationto get Θ M → ∧ Θ M → · · · ∧ n Θ M → since it looks more natural by the general philosophy ofdeformation theory so that the -th cohomology group corresponds to infinitesimal Poisson auto-morphisms, the first cohomology group corresponds to infinitesimal Poisson deformations and thethird cohomology group corresponds to obstructions ( see the third part of the author’s Ph.D. thesis [Kim14]) . We will relate the first ‘degree-shifted by 1’ truncated holomorphic Poisson cohomology group H ( M t , Θ • M t ) to infinitesimal deformations of ω − ( t ) = ( M t , Λ t ) in a Poisson analytic family( M , Λ , B, ω ) for each t ∈ B . As in Remark 2 . .
8, let U = {U j } be an open covering of M such that U j are coordinate polydisks of M , { ( z j , t ) } = { ( z j , ..., z nj , t , ..., t m ) } is a local complexcoordinate system on U j , and z αj = f αjk ( z k , ..., z nk , t , ..., t m ) , α = 1 , ..., n is a holomorphic transitionfunction from z k to z j . The Poisson structure Λ is expressed in terms of local complex coordinatesystem on U j as Λ = Λ j = n X α,β =1 g jαβ ( z j , t ) ∂∂z αj ∧ ∂∂z βj (3.1.4)where g jαβ ( z j , t ) is a holomorphic function on U j with g jαβ ( z j , t ) = − g jβα ( z j , t ) and we have[Λ , Λ] = [ n X α,β =1 g jαβ ( z j , t ) ∂∂z αj ∧ ∂∂z βj , n X α,β =1 g jαβ ( z j , t ) ∂∂z αj ∧ ∂∂z βj ] = 0(3.1.5)Since f jk ( z k , t ) = ( f jk ( z k , t ) , ..., f njk ( z k , t ) , t , ..., t m ) is a Poisson map, we have g jαβ ( f jk ( z k , t ) , ..., f njk ( z k , t )) = n X r,s =1 g krs ( z k , t ) ∂f αjk ∂z rk ∂f βjk ∂z sk (3.1.6)on U j ∩ U k . Set U tj := U j ∩ M t . Then for each t ∈ B , U t := {U tj } is an open covering of M t . Recallthat Λ t is the Poisson structure on M t induced from ( M , Λ). Let ∂∂t = P mλ =1 c λ ∂∂t λ , c λ ∈ C be atangent vector of B . Then we have Proposition 3.1.7. ( { λ j ( t ) = n X α,β =1 ∂g jαβ ( z j , t ) ∂t ∂∂z αj ∧ ∂∂z βj } , { θ jk ( t ) = n X α =1 ∂f αjk ( z k , t ) ∂t ∂∂z αj } ) ∈ C ( U t , ∧ Θ M t ) ⊕ C ( U t , Θ M t ) define a 1-cocycle and call its cohomology class in H ( M t , Θ • M t ) the infinitesimal ( Poisson ) defor-mation along ∂∂t . This expression is independent of the choice of system of local coordinates.Proof. First we note that δ ( { θ jk ( t ) } ) = 0 (See [Kod05] p.201). Second, by taking the derivative of(3 . .
5) with respect to t , we have [ P nα,β =1 g jαβ ( z j , t ) ∂∂z αj ∧ ∂∂z βj , P nα,β =1 ∂g jαβ ( z j ,t ) ∂t ∂∂z αj ∧ ∂∂z βj ] = 0. Itremains to show that δ ( { λ j ( t ) } ) + [Λ t , { θ jk } ] = 0. More precisely, on U tj ∩ U tk = ∅ , we show that λ k ( t ) − λ j ( t ) + [Λ t , θ jk ( t )] = 0. In other words, n X r,s =1 ∂g krs ∂t ∂∂z rk ∧ ∂∂z sk − n X α,β =1 ∂g jαβ ∂t ∂∂z αj ∧ ∂∂z βj + [ n X r,s =1 g jrs ( z j , t ) ∂∂z rj ∧ ∂∂z sj , n X c =1 ∂f cjk ( z k , t ) ∂t ∂∂z cj ] = 0(3.1.8) HEOREM OF EXISTENCE AND COMPLETENESS FOR HOLOMORPHIC POISSON STRUCTURES 7
Since z αj = f αjk ( z k , ..., z nk , t , ..., t m ) for α = 1 , ..., n , we have ∂∂z rk = P na =1 ∂f ajk ∂z rk ∂∂z aj for r = 1 , ..., n .Hence the first term of (3 . .
8) is n X r,s =1 ∂g krs ∂t ∂∂z rk ∧ ∂∂z sk = n X r,s,a,b =1 ∂g krs ∂t ∂f ajk ∂z rk ∂f bjk ∂z sk ∂∂z aj ∧ ∂∂z bj We compute the third term of (3 . . n X r,s,c =1 [ g jrs ( z, t ) ∂∂z rj ∧ ∂∂z sj , ∂f cjk ( z k , t ) ∂t ∂∂z cj ] = n X r,s,c =1 ([ g jrs ∂∂z rj , ∂f cjk ∂t ∂∂z cj ] ∧ ∂∂z sj − g jrs [ ∂∂z sj , ∂f cjk ∂t ∂∂z cj ] ∧ ∂∂z rj )= n X r,s,c =1 ( g jrs ∂∂z rj (cid:18) ∂f cjk ∂t (cid:19) ∂∂z cj ∧ ∂∂z sj − ∂f cjk ∂t ∂g jrs ∂z cj ∂∂z rj ∧ ∂∂z sj + g jrs ∂∂z sj (cid:18) ∂f cjk ∂t (cid:19) ∂∂z rj ∧ ∂∂z cj )By considering the coefficients of ∂∂z aj ∧ ∂∂z bj , (3 . .
8) is equivalent to n X r,s =1 ∂g krs ∂t ∂f ajk ∂z rk ∂f bjk ∂z sk − ∂g jab ∂t − n X c =1 ∂g jab ∂z cj ∂f cjk ∂t + n X c =1 ( g jcb ∂∂z cj (cid:18) ∂f ajk ∂t (cid:19) + g jac ∂∂z cj ∂f bjk ∂t ! ) = 0(3.1.9)On the other hand, from (3 . . g jab ( f jk ( z k , t ) , ..., f njk ( z k , t ) , t , ..., t m ) = n X r,s =1 g krs ∂f ajk ∂z rk ∂f bjk ∂z sk on U j ∩ U k (3.1.10)By taking the derivative of (3 . .
10) with respect to t , we have n X c =1 ∂g jab ∂z cj ∂f cjk ∂t + ∂g jab ∂t = n X r,s =1 ∂g krs ∂t ∂f ajk ∂z rk ∂f bjk ∂z sk + n X r,s =1 g krs ( ∂∂z rk (cid:18) ∂f ajk ∂t (cid:19) ∂f bjk ∂z sk + ∂f ajk ∂z rk ∂∂z sk ∂f bjk ∂t ! )Hence (3 . .
9) is equivalent to n X c =1 ( g jcb ∂∂z cj (cid:18) ∂f ajk ∂t (cid:19) + g jac ∂∂z cj ∂f bjk ∂t ! ) = n X r,s =1 g krs ( ∂∂z rk (cid:18) ∂f ajk ∂t (cid:19) ∂f bjk ∂z sk + ∂f ajk ∂z rk ∂∂z sk ∂f bjk ∂t ! )(3.1.11)Indeed, the left hand side and right hand side of (3 . .
11) coincide: from (3 . . n X c =1 ( g jcb ∂∂z cj (cid:18) ∂f ajk ∂t (cid:19) + g jac ∂∂z cj ∂f bjk ∂t ! ) = n X r,s,c =1 ( g krs ∂f cjk ∂z rk ∂f bjk ∂z sk ∂∂z cj (cid:18) ∂f ajk ∂t (cid:19) + g krs ∂f ajk ∂z rk ∂f cjk ∂z sk ∂∂z cj ∂f bjk ∂t ! )= n X r,s =1 g krs ( ∂∂z rk (cid:18) ∂f ajk ∂t (cid:19) ∂f bjk ∂z sk + ∂f ajk ∂z rk ∂∂z sk ∂f bjk ∂t ! ) This proves the first claim. It remains to show that ( { λ j ( t ) } , { θ jk ( t ) } ) is independent of the choice ofsystems of local coordinates. We can show that the infinitesimal deformation does not change underthe refinement of the open covering (See [Kod05] p.190). Since we can choose a common refinementfor two system of local coordinates, it is sufficient to show that given two local coordinates x j =( z j , t ) and u j = ( w j , t ) on each U j , the infinitesimal Poisson deformation ( { π j ( t ) } , { η jk ( t ) } ) withrespect to { u j } coincides with ( { λ j ( t ) } , { θ jk ( t ) } ) with respect to { x j } . Let the Poisson structure Λin (3.1.4) be expressed in terms of local coordinates u j as Λ = Π j = P nα,β =1 Π jαβ ( w j , t ) ∂∂w αj ∧ ∂∂w βj .Let ( w k , t ) → ( w j , t ) = ( e jk ( w k , t ) , t ) be the coordinate transformation of { u j } on U j ∩ U k = ∅ . Nowwe set η jk ( t ) = n X α =1 ∂e αjk ( w k , t ) ∂t ∂∂w αj , w k = e kj ( w j , t ) , π j ( t ) = n X α,β =1 ∂ Π jαβ ( w j , t ) ∂t ∂∂w αj ∧ ∂∂w βj CHUNGHOON KIM
We show that ( { λ j ( t ) } ) , { θ jk ( t ) } ) is cohomologous to ( { π j ( t ) } , { η jk ( t ) } ). Let w αj = h αj ( z j , ..., z nj , t ) , α =1 , ..., n , define the coordinate transformation from x j = ( z j , t ) to u j = ( w j , t ) which is a Poissonmap. So we have ∂∂z rj = P na =1 ∂h aj ∂z rj ∂∂w aj and the following relation holdsΠ jαβ ( h j ( z j , t ) , ..., h nj ( z j , t ) , t ) = n X r,s =1 g jrs ( z j , t ) ∂h αj ∂z rj ∂h βj ∂z sj . (3.1.12)Set θ j ( t ) = P nα =1 ∂h αj ( z j ,t ) ∂t ∂∂w αj , w αj = h αj ( z j , t ). Then we claim that ( λ j ( t ) , θ jk ( t )) − ( π j ( t ) , η jk ( t )) = θ k ( t ) − θ j ( t ) − [Λ t , θ j ( t )] = − δ ( − θ j ( t ))+[Λ t , − θ j ( t )], which means ( { λ j ( t ) } , { θ jk ( t ) } ) is cohomologousto ( { π j ( t ) } , { η jk ( t ) } ). Since δ ( { θ j ( t ) } ) = { θ jk ( t ) } − { η jk ( t ) } (for the detail, see [Kod05] p.191-192),we only need to see λ j ( t ) − π j ( t ) + [Λ t (= Π t ) , θ j ( t )] = 0. Equivalently, n X r,s =1 ∂g jrs ( z j , t ) ∂t ∂∂z rj ∧ ∂∂z sj − n X α,β =1 ∂ Π jαβ ( w j , t ) ∂t ∂∂w αj ∧ ∂∂w βj + [ n X α,β =1 Π jαβ ( w j , t ) ∂∂w αj ∧ ∂∂w βj , n X c =1 ∂h cj ( z j , t ) ∂t ∂∂w cj ] = 0 which follows from taking the derivative (3.1.12) with respect to t as in the proof of the firstclaim. (cid:3) Definition 3.1.13 ((holomorphic) Poisson Kodaira-Spencer map) . Let ( M , Λ , B, ω ) be a Poissonanalytic family, where B is a domain of C m . As in Remark . . , let U = {U j } be an opencovering of M , and ( z j , t ) a local complex coordinate system on U j . The Poisson structure Λ is expressed as P nα,β =1 g jαβ ( z j , t ) ∂∂z αj ∧ ∂∂z βj on U j where g jαβ ( z j , t ) is a holomorphic function with g jαβ ( z j , t ) = − g jβα ( z j , t ) . For a tangent vector ∂∂t = P mλ =1 c λ ∂∂t λ , c λ ∈ C , of B , we put ∂ Λ t ∂t := n X α,β =1 " m X λ =1 c λ ∂g jαβ ( z j , t ) ∂t λ ∂∂z αj ∧ ∂∂z βj The ( holomorphic ) Poisson Kodaira-Spencer map is defined to be a C -linear map ϕ t : T t ( B ) → H ( M t , Θ • M t ) ∂∂t (cid:20) ρ t (cid:18) ∂∂t (cid:19) (cid:18) = ∂M t ∂t (cid:19) , ∂ Λ t ∂t (cid:21) = ∂ ( M t , Λ t ) ∂t where ρ t : T t ( B ) → H ( M t , Θ t ) is the Kodaira-Spencer map of the complex analytic family ( M , B, ω )( see [Kod05] p. . Integrability condition
In a complex analytic family ( M , B, ω ) of deformations of a complex manifold M = ω − (0), thedeformations near M are represented by C ∞ vector (1 , ϕ ( t ) ∈ A , ( M, T M ) on M satisfying ϕ (0) = 0 and the integrability condition ¯ ∂ϕ ( t ) − [ ϕ ( t ) , ϕ ( t )] = 0 where t ∈ ∆ a sufficiently smallpolydisk in B (see [Kod05] section § M , B, Λ , ω ) of deformations of a compact holomorphic Poisson manifold ( M, Λ ) = ω − (0), thedeformations near ( M, Λ ) are represented by C ∞ vector (0 , ϕ ( t ) ∈ A , ( M, T M ) and C ∞ bivectors Λ( t ) ∈ A , ( M, ∧ T M ) satisfying ϕ (0) = 0, Λ(0) = Λ and the integrability condition¯ ∂ ( ϕ ( t ) + Λ( t )) + [ ϕ ( t ) + Λ( t ) , ϕ ( t ) + Λ( t )] = 0. To deduce the integrability condition, we extendKodaira’s approach ([Kod05] section § Preliminaries.
We extend the argument of [Kod05] p.259-261 (to which we refer for the detail) in the contextof a Poisson analytic family. We tried to maintain notational consistency with [Kod05].Let ( M , Λ , B, ω ) be a Poisson analytic family of compact Poisson holomorphic manifolds, where B is a domain of C m containing the origin 0. Define | t | = max λ | t λ | for t = ( t , ..., t m ) ∈ C m , andlet ∆ = ∆ r = { t ∈ C m || t | < r } the polydisk of radius r >
0. If we take a sufficiently small ∆ ⊂ B ,then ( M ∆ , Λ ∆ ) = ω − (∆) is represented in the form( M ∆ , Λ ∆ ) = [ j ( U j × ∆ , Λ | U j × ∆ ) HEOREM OF EXISTENCE AND COMPLETENESS FOR HOLOMORPHIC POISSON STRUCTURES 9
We denote a point of U j by ξ j = ( ξ j , ..., ξ nj ) and its holomorphic Poisson structure Λ | U j × ∆ by P nα,β =1 g jαβ ( ξ j , t ) ∂∂ξ αj ∧ ∂∂ξ βj on U j × ∆ with g jαβ ( ξ j , t ) = − g jβα ( ξ j , t ). For simplicity, we assume that U j = { ξ j ∈ C m || ξ j | < } where | ξ | = max a | ξ aj | . ( ξ j , t ) ∈ U j × ∆ and ( ξ k , t ) ∈ U k × ∆ are thesame point on M ∆ if ξ αj = f αjk ( ξ k , t ), α = 1 , ..., n where f jk ( ξ k , t ) is a Poisson holomorphic map of ξ k , ..., ξ nk , t , ..., t m , defined on U k × ∆ ∩ U j × ∆, and so we have the following relation g jαβ ( f jk ( ξ k , t ) , ..., f njk ( ξ k , t )) = n X r,s =1 g krs ( ξ k , t ) ∂f αjk ∂ξ rk ∂f βjk ∂ξ sk (4.1.1)We note that ω − ( t ) = ( M t , Λ t ) = S j ( U j , P nα,β =1 g jαβ ( ξ j , t ) ∂∂ξ αj ∧ ∂∂ξ βj ) for t ∈ ∆By [Kod05] Theorem 2.3, when we ignore complex structures and Poisson structures, M t isdiffeomorphic to M = ω − (0) as differentiable manifolds for each t ∈ ∆. We put M := M . By[Kod05] Theorem 2.5, if we take a sufficiently small ∆, there is a diffeomorphism Ψ of M × ∆ onto M ∆ as differentiable manifolds such that ω ◦ Ψ is the projection M × ∆ → ∆. Let z = ( z , ..., z n ) belocal complex coordinates of M = M . Then we have ω ◦ Ψ( z, t ) = t, t ∈ ∆. For Ψ( z, t ) ∈ U j × ∆,put Ψ( z, t ) = ( ξ j ( z, t ) , ..., ξ nj ( z, t ) , t , ..., t m ) . (4.1.2)Then each component ξ αj = ξ αj ( z, t ), α = 1 , ..., n is a C ∞ function. If we identify M ∆ = Ψ( M × ∆)with M × ∆ via Ψ, ( M ∆ , Λ ∆ ) is considered as a holomorphic Poisson manifold with the complexstructure defined on the C ∞ manifold M × ∆ by the system of local coordinates on U j × ∆ { ( ξ j , t ) | j = 1 , , , ... } , ( ξ j , t ) = ( ξ j ( z, t ) , ..., ξ nj ( z, t ) , t , ..., t m ) . and the holomorphic Poisson structure given by on U j × ∆ { n X α,β =1 g jαβ ( ξ j ( z, t ) , t ) ∂∂ξ αj ∧ ∂∂ξ βj | j = 1 , , , ... } (4.1.3)We note that since ( z , ..., z n ) and ( ξ j ( z, , ..., ξ nj ( z, M = M , ξ αj ( z,
0) are holomorphic functions of z , ..., z n , α = 1 , ..., n (4.1.4)We also note that if we take ∆ sufficiently small, we havedet (cid:18) ∂ξ αj ( z, t ) ∂z λ (cid:19) α,λ =1 ,...,n = 0(4.1.5)for any t ∈ ∆.With this preparation, we identify the holomorphic Poisson deformations near ( M, Λ ) in thePoisson analytic family ( M , Λ , B, ω ) with ϕ ( t ) + Λ( t ) where ϕ ( t ) is a C ∞ vector (0 , t ) is a C ∞ bivector on M for t ∈ ∆.4.2. Identification of the deformations of complex structures with ϕ ( t ) ∈ A , ( M, T M ) . Put U j = Ψ − ( U j × ∆). Then U j ⊂ M × ∆ is the domain of ξ αj ( z, t ). From (4 . . , ϕ λj ( z, t ) = P nv =1 ϕ λjv ( z, t ) d ¯ z v in the following way: ϕ j ( z, t )... ϕ nj ( z, t ) := ∂ξ j ∂z . . . ∂ξ j ∂z n ... ... ∂ξ nj ∂z . . . ∂ξ nj ∂z n − ¯ ∂ξ j ...¯ ∂ξ nj Then the coefficients ϕ αjv ( z, t ) are C ∞ functions on U j and ¯ ∂ξ αj ( z, t ) = P nλ =1 ϕ λj ( z, t ) ∂ξ αj ( z,t ) ∂z λ , α =1 , ..., n . So we have(4.2.1) ∂ξ αj ∂ ¯ z v = n X λ =1 ϕ λjv ( z, t ) ∂ξ αj ∂z λ Lemma 4.2.2. On U j ∩ U k , we have n X λ =1 ϕ λj ( z, t ) ∂∂z λ = n X λ =1 ϕ λk ( z, t ) ∂∂z λ Proof.
See [Kod05] p.262. (cid:3)
If for ( z, t ) ∈ U j , we define ϕ ( z, t ) := n X λ =1 ϕ λj ( z, t ) ∂∂z λ = n X λ =1 ϕ λ ( z, t ) ∂∂z λ = n X v,λ =1 ϕ λv ( z, t ) d ¯ z v ∂∂z λ (4.2.3)By Lemma 4.2.2, ϕ ( t ) = ϕ ( z, t ) ∈ A , ( M, T M ) is a C ∞ vector (0 , M for every t ∈ ∆and we have ϕ (0) = 0, ¯ ∂ϕ ( t ) −
12 [ ϕ ( t ) , ϕ ( t )] = 0(4.2.4)(see [Kod05] p.263,p.265). We also point out that Theorem 4.2.5.
If we take a sufficiently small polydisk ∆ as in subsection . , then for t ∈ ∆ , alocal C ∞ function f on M is holomorphic with respect to the complex structure M t if and only if f satisfies the equation ( ¯ ∂ − ϕ ( t )) f = 0 Proof.
See [Kod05] Theorem 5.3 p.263. (cid:3)
Identification of the deformations of Poisson structures with Λ( t ) ∈ A , ( M, ∧ T M ) . For the holomorphic Poisson structure P nα,β =1 g jαβ ( ξ j ( z, t ) , t ) ∂∂ξ βj ∧ ∂∂ξ βj on each U j × ∆ from (4 . . j ( z, t ) := P nr,s =1 h jrs ( z, t ) ∂∂z r ∧ ∂∂z s on U j = Ψ − ( U j × ∆) suchthat n X r,s =1 h jrs ( z, t ) ∂ξ αj ∂z r ∂ξ βj ∂z s = g jαβ ( ξ j ( z, t ) , t ) . (4.3.1)Indeed, from (4 . . h j ( z, t ) . . . h j n ( z, t )... ... ... h jn ( z, t ) . . . h jnn ( z, t ) := ∂ξ j∂z . . . ∂ξ j∂zn ... ... ... ∂ξnj∂z . . . ∂ξnj∂zn − g j ( ξ j ( z, t )) . . . g j n ( ξ j ( z, t ))... ... ... g jn ( ξ j ( z, t )) . . . g jnn ( ξ j ( z, t )) ∂ξ j∂z . . . ∂ξnj∂z ... ... ... ∂ξ j∂zn . . . ∂ξnj∂zn − We note that since g jαβ ( ξ j ( z, t )) = − g jβα ( ξ j ( z, t )), we have h jrs ( z, t ) = − h jsr ( z, t ). Lemma 4.3.2. On U j ∩ U k , we have h jrs ( z, t ) = h krs ( z, t ) .Proof. From (4 . . . .
1) and ∂ξ αj ∂z r = P np =1 ∂ξ pk ∂z r ∂ξ αj ∂ξ pk , we have n X r,s =1 h jrs ( z, t ) ∂ξ αj ∂z r ∂ξ βj ∂z s = g jαβ ( ξ j ( z, t ) , t ) = n X p,q =1 g kpq ( ξ k ( z, t ) , t ) ∂ξ αj ∂ξ pk ∂ξ βj ∂ξ qk = n X p,q,r,s =1 h krs ( z, t ) ∂ξ pk ∂z r ∂ξ qk ∂z s ∂ξ αj ∂ξ pk ∂ξ βj ∂ξ qk = n X r,s =1 h krs ( z, t ) ∂ξ αj ∂z r ∂ξ βj ∂z s . From (4 . . h jrs ( z, t ) = h krs ( z, t ). (cid:3) If for ( z, t ) ∈ U j , we defineΛ( z, t ) := n X r,s =1 h jrs ( z, t ) ∂∂z r ∧ ∂∂z s = n X r,s =1 h rs ( z, t ) ∂∂z r ∧ ∂∂z s . (4.3.3)By Lemma 4.3.2, Λ( t ) := Λ( z, t ) ∈ A , ( M, ∧ T M ) is a C ∞ bivector field on M for every t ∈ ∆with Λ(0) = Λ . HEOREM OF EXISTENCE AND COMPLETENESS FOR HOLOMORPHIC POISSON STRUCTURES 11
Theorem 4.3.4.
If we take a sufficiently small polydisk ∆ as in subsection . , then for thePoisson structure P nα,β =1 g jαβ ( ξ j , t ) ∂∂ξ βj ∧ ∂∂ξ βj on U j × ∆ for each j , there exists the unique bivectorfield Λ j ( t ) = P nr,s =1 h jrs ( z, t ) ∂∂z r ∧ ∂∂z s on U j satisfying (1) P nr,s =1 h jrs ( z, t ) ∂ξ αj ∂z r ∂ξ βj ∂z s = g jαβ ( ξ j ( z, t ) , t )(2) Λ j ( t ) are glued together to define a C ∞ bivector field Λ( t ) on M × ∆(3) for each j , [Λ j ( t ) , Λ j ( t )] = 0 . Hence we have [Λ( t ) , Λ( t )] = 0We will use the following lemma to prove the theorem. Lemma 4.3.5. If σ = P nα,β =1 σ αβ ∂∂z α ∧ ∂∂z β with σ αβ = − σ βα , then [ σ, σ ] = 0 is equivalent to n X l =1 ( σ lk ∂σ ij ∂z l + σ li ∂σ jk ∂z l + σ lj ∂σ ki ∂z l ) = 0 for each ≤ i, j, k ≤ n .Proof of Theorem . . . We have already showed (1) and (2). It remains to show (3). We notethat [ n X α,β =1 g jαβ ( ξ j , t ) ∂∂ξ αj ∧ ∂∂ξ βj , n X α,β =1 g jαβ ( ξ j , t ) ∂∂ξ αj ∧ ∂∂ξ βj ] = 0 . (4.3.6)Since g jαβ ( ξ j ( z, t ) , t ) = P na,b =1 h jab ( z, t ) ∂ξ αj ∂z a ∂ξ βj ∂z b is holomorphic with respect to ξ j = ( ξ αj ) , α = 1 , ..., n ,we have ∂ ¯ ∂ξ αj n X a,b =1 h jab ( z, t ) ∂ξ αj ∂z a ∂ξ βj ∂z b = n X a,b =1 ∂ ¯ ∂ξ αj h jab ( z, t ) ∂ξ αj ∂z a ∂ξ βj ∂z b ! = 0(4.3.7)In the following, for simplicity, we denote ξ αj ( z j , t ) by ξ α and h jab ( z, t ) by h ab . By (4.3.6), Lemma4.3.5 and (4 . . h ab = − h ba and ∂∂z a = P nl =1 ∂ξ l ∂z a ∂∂ξ l + P nl =1 ∂ ¯ ξ l ∂z a ∂∂ ¯ ξ l , we have n X a,b,c,d,l =1 ( h ab ∂ξ l ∂z a ∂ξ k ∂z b ∂∂ξ l (cid:18) h cd ∂ξ i ∂z c ∂ξ j ∂z d (cid:19) + h ab ∂ξ l ∂z a ∂ξ i ∂z b ∂∂ξ l (cid:18) h cd ∂ξ j ∂z c ∂ξ k ∂z d (cid:19) + h ab ∂ξ l ∂z a ∂ξ j ∂z b ∂∂ξ l (cid:18) h cd ∂ξ k ∂z c ∂ξ i ∂z d (cid:19) )+ n X a,b,c,d,l =1 ( h ab ∂ ¯ ξ l ∂z a ∂ξ k ∂z b ∂∂ ¯ ξ l (cid:18) h cd ∂ξ i ∂z c ∂ξ j ∂z d (cid:19) + h ab ∂ ¯ ξ l ∂z a ∂ξ i ∂z b ∂∂ ¯ ξ l (cid:18) h cd ∂ξ j ∂z c ∂ξ k ∂z d (cid:19) + h ab ∂ ¯ ξ l ∂z a ∂ξ j ∂z b ∂∂ ¯ ξ l (cid:18) h cd ∂ξ k ∂z c ∂ξ i ∂z d (cid:19) )= n X a,b,c,d,l =1 ( h ab ∂ξ l ∂z a ∂ξ k ∂z b ∂h cd ∂ξ l ∂ξ i ∂z c ∂ξ j ∂z d + h ab ∂ξ l ∂z a ∂ξ k ∂z b h cd ∂∂ξ l (cid:18) ∂ξ i ∂z c (cid:19) ∂ξ j ∂z d + h ab ∂ξ l ∂z a ∂ξ k ∂z b h cd ∂ξ i ∂z c ∂∂ξ l (cid:18) ∂ξ j ∂z d (cid:19) )+ n X a,b,c,d,l =1 ( h ab ∂ξ l ∂z a ∂ξ i ∂z b ∂h cd ∂ξ l ∂ξ j ∂z c ∂ξ k ∂z d + h ab ∂ξ l ∂z a ∂ξ i ∂z b h cd ∂∂ξ l (cid:18) ∂ξ j ∂z c (cid:19) ∂ξ k ∂z d + h ab ∂ξ l ∂z a ∂ξ i ∂z b h cd ∂ξ j ∂z c ∂∂ξ l (cid:18) ∂ξ k ∂z d (cid:19) )+ n X a,b,c,d,l =1 ( h ab ∂ξ l ∂z a ∂ξ j ∂z b ∂h cd ∂ξ l ∂ξ k ∂z c ∂ξ i ∂z d + h ab ∂ξ l ∂z a ∂ξ j ∂z b h cd ∂∂ξ l (cid:18) ∂ξ k ∂z c (cid:19) ∂ξ i ∂z d + h ab ∂ξ l ∂z a ∂ξ j ∂z b h cd ∂ξ k ∂z c ∂∂ξ l (cid:18) ∂ξ i ∂z d (cid:19) )+ n X a,b,c,d,l =1 ( h ab ∂ ¯ ξ l ∂z a ∂ξ k ∂z b ∂h cd ∂ ¯ ξ l ∂ξ i ∂z c ∂ξ j ∂z d + h ab ∂ ¯ ξ l ∂z a ∂ξ k ∂z b h cd ∂∂ ¯ ξ l (cid:18) ∂ξ i ∂z c (cid:19) ∂ξ j ∂z d + h ab ∂ ¯ ξ l ∂z a ∂ξ k ∂z b h cd ∂ξ i ∂z c ∂∂ ¯ ξ l (cid:18) ∂ξ j ∂z d (cid:19) )+ n X a,b,c,d,l =1 ( h ab ∂ ¯ ξ l ∂z a ∂ξ i ∂z b ∂h cd ∂ ¯ ξ l ∂ξ j ∂z c ∂ξ k ∂z d + h ab ∂ ¯ ξ l ∂z a ∂ξ i ∂z b h cd ∂∂ ¯ ξ l (cid:18) ∂ξ j ∂z c (cid:19) ∂ξ k ∂z d + h ab ∂ ¯ ξ l ∂z a ∂ξ i ∂z b h cd ∂ξ j ∂z c ∂∂ ¯ ξ l (cid:18) ∂ξ k ∂z d (cid:19) )+ n X a,b,c,d,l =1 ( h ab ∂ ¯ ξ l ∂z a ∂ξ j ∂z b ∂h cd ∂ ¯ ξ l ∂ξ k ∂z c ∂ξ i ∂z d + h ab ∂ ¯ ξ l ∂z a ∂ξ j ∂z b h cd ∂∂ ¯ ξ l (cid:18) ∂ξ k ∂z c (cid:19) ∂ξ i ∂z d + h ab ∂ ¯ ξ l ∂z a ∂ξ j ∂z b h cd ∂ξ k ∂z c ∂∂ ¯ ξ l (cid:18) ∂ξ i ∂z d (cid:19) )= n X a,b,c,d =1 ( h ab ∂h cd ∂z a ∂ξ k ∂z b ∂ξ i ∂z c ∂ξ j ∂z d + h ab ∂ξ k ∂z b h cd ∂ ξ i ∂z a ∂z c ∂ξ j ∂z d + h ab ∂ξ k ∂z b h cd ∂ξ i ∂z c ∂ ξ j ∂z a ∂z d )+ n X a,b,c,d =1 ( h ab ∂h cd ∂z a ∂ξ i ∂z b ∂ξ j ∂z c ∂ξ k ∂z d + h ab ∂ξ i ∂z b h cd ∂ ξ j ∂z a ∂z c ∂ξ k ∂z d + h ab ∂ξ i ∂z b h cd ∂ξ j ∂z c ∂ ξ k ∂z a ∂z d )+ n X a,b,c,d =1 ( h ab ∂h cd ∂z a ∂ξ j ∂z b ∂ξ k ∂z c ∂ξ i ∂z d + h ab ∂ξ j ∂z b h cd ∂ ξ k ∂z a ∂z c ∂ξ i ∂z d + h ab ∂ξ j ∂z b h cd ∂ξ k ∂z c ∂ ξ i ∂z a ∂z d )= n X a,b,c,d =1 ( h ab ∂h cd ∂z a ∂ξ k ∂z b ∂ξ i ∂z c ∂ξ j ∂z d + h ab ∂h cd ∂z a ∂ξ i ∂z b ∂ξ j ∂z c ∂ξ k ∂z d + h ab ∂h cd ∂z a ∂ξ j ∂z b ∂ξ k ∂z c ∂ξ i ∂z d )= n X a,b,c,d =1 (cid:18) h ab ∂h cd ∂z a + h ac ∂h db ∂z a + h ad ∂h bc ∂z a (cid:19) ∂ξ i ∂z c ∂ξ j ∂z d ∂ξ k ∂z b From (4 . . P na =1 h ab ∂h cd ∂z a + h ac ∂h db ∂z a + h ad ∂h bc ∂z a = 0 for each b, c, d . So by Lemma 4.3.5,[Λ j ( t ) , Λ j ( t )] = 0. (cid:3) Remark 4.3.8.
For the compact holomorphic Poisson manifold ( M t , Λ t ) for each t ∈ ∆ in thePoisson analytic family ( M , Λ , B, ω ) , we showed that there exists a bivector field Λ( t ) on M = M with [Λ( t ) , Λ( t )] = 0 for t ∈ ∆ by Theorem . . . Let J t : T R M → T R M with J t = − id be the almostcomplex structure associated to the complex structure M t ( induced by ϕ ( t )) where T R M is a realtangent bundle of the underlying differentiable manifold M . Then J t induces a type decompositionof complexified tangent bundle T C M = T , M t ⊕ T , M t ( see [KN96] Chapter IX section 2 ) so that wehave ∧ T C M = ∧ T , M t ⊕ T , M t ⊗ T , M t ⊕∧ T , M t . If Λ is a C ∞ section of ∧ T C M on M , then we denoteby Λ , the component of ∧ T , M t , by Λ , the component of T , M t ⊗ T , M t , and by Λ , the componentof ∧ T , M t . So we have Λ = Λ , + Λ , + Λ , . We call Λ , the type (2 , -part of Λ . With thisnotation, the type (2 , -part of Ψ ∗ Λ( t ) is Λ t for t ∈ ∆ , where Ψ ∗ Λ( t ) is the bivector field inducedfrom Λ( t ) via diffeomorphism Ψ in (4 . . . So we can say that Λ( t ) , = Λ t . HEOREM OF EXISTENCE AND COMPLETENESS FOR HOLOMORPHIC POISSON STRUCTURES 13
Remark 4.3.9.
Let Λ be a C ∞ -section of ∧ k T C M . From ∧ k T C M = L p + q = k ∧ p T , M t ⊗ ∧ q T , M t , wecan define the type ( p, q ) part Λ p,q of Λ in an obvious way as in Remark . . . Next we discuss the condition when a given C ∞ bivector field Λ ∈ A , ( M, ∧ T M ) on M with[Λ , Λ] = 0 gives a holomorphic bivector field Λ , ∈ A , ( M t , ∧ T M t ) with respect to the complexstructure M t induced by ϕ ( t ). Before proceeding our discussion, we recall the Schouten bracket[ − , − ] on L i ≥ L p + q − i,p ≥ ,q ≥ A ,p ( M, ∧ q T M ) (see (1.0.1)) which we need for the computationof the integrability condition (4.5.5). The Schouten bracket [ − , − ] is defined in the following way:[ − , − ] : A ,p ( M, ∧ q T M ) × A ,p ′ ( M, ∧ q ′ T M ) → A ,p + p ′ ( M, ∧ q + q ′ − T M )In local coordinates it is given by[ f d ¯ z I ∂∂z J , gd ¯ z K ∂∂z L ] = ( − | K | ( | J | +1) d ¯ z I ∧ d ¯ z K [ f ∂∂z J , g ∂∂z L ](4.3.10)where f, g are C ∞ functions on M and d ¯ z I = d ¯ z i ∧ · · · ∧ d ¯ z i | I | , ∂∂z J = ∂∂z j ∧ · · · ∧ ∂∂z j | L | (similarlyfor d ¯ z K , ∂∂z L ). Then g = ( M i ≥ g i , g i = M p + q − i,p ≥ ,q ≥ A ,p ( M, ∧ q T M ) , L = ¯ ∂ + [Λ , − ] , [ − , − ]) , (4.3.11)is a differential graded Lie algebra. So we have the following properties: for a ∈ A ,p ( M, ∧ q T M ) , b ∈ A ,p ′ ( M, ∧ q ′ T M ), and c ∈ A ,p ′′ ( M, ∧ q ′′ T M )(1) [ a, b ] = − ( − ( p + q +1)( p ′ + q ′ +1) [ b, a ](2) [ a, [ b, c ]] = [[ a, b ] , c ] + ( − ( p + q +1)( p ′ + q ′ +1) [ b, [ a, c ]](3) ¯ ∂ [ a, b ] = [ ¯ ∂a, b ] + ( − p + q +1 [ a, ¯ ∂b ] Theorem 4.3.12.
If we take a sufficiently small polydisk ∆ as in subsection . , then for t ∈ ∆ ,a type (2 , -part Λ , of a C ∞ bivector field Λ = P nr,s =1 h rs ( z ) ∂∂z r ∧ ∂∂z s on M is holomorphic withrespect to the complex structure M t induced by ϕ ( t ) if and only if it satisfies the equation ¯ ∂ Λ − [Λ , ϕ ( t )] = 0 Moreover, if [Λ , Λ] = 0 , then [Λ , , Λ , ] = 0 .Proof. We note that the type (2 , P nr,s =1 h rs ( z ) ∂∂z r ∧ ∂∂z s with respect to complexstructure M t is P nr,s,α,β =1 h rs ∂ξ αj ∂z r ∂ξ βj ∂z s ∂∂ξ αj ∧ ∂∂ξ βj . Hence by Theorem 4.2.5, it suffices to show thatFor each α, β , ( ¯ ∂ − ϕ ( t ))( n X r,s =1 h rs ∂ξ αj ∂z r ∂ξ βj ∂z s ) = 0 if and only if ¯ ∂ Λ − [Λ , ϕ ( t )] = 0(4.3.13)First we note that from (4 . .
3) and (4 . . ∂ Λ − [Λ , ϕ ( t )](4.3.14)= n X r,s,v =1 ∂h rs ∂ ¯ z v d ¯ z v ∂∂z r ∧ ∂∂z s − n X r,s,v,λ =1 [ h rs ∂∂z r ∧ ∂∂z s , ϕ λv d ¯ z v ∂∂z λ ]= n X r,s,v =1 ∂h rs ∂ ¯ z v d ¯ z v ∂∂z r ∧ ∂∂z s + n X r,s,v,λ =1 [ h rs ∂∂z r ∧ ∂∂z s , ϕ λv ∂∂z λ ] d ¯ z v = n X r,s,v =1 ∂h rs ∂ ¯ z v d ¯ z v ∂∂z r ∧ ∂∂z s + n X r,s,v,λ =1 ( h rs ∂φ λv ∂z r ∂∂z λ ∧ ∂∂z s − ϕ λv ∂h rs ∂z λ ∂∂z r ∧ ∂∂z s + h rs ∂ϕ λv ∂z s ∂∂z r ∧ ∂∂z λ ) d ¯ z v . By considering the coefficients of d ¯ z v ∂∂z r ∧ ∂∂z s in (4 . . ∂ Λ − [Λ , ϕ ( t )] = 0 is equivalent to ∂h rs ∂ ¯ z v + n X c =1 ( h cs ∂ϕ rv ∂z c − ϕ cv ∂h rs ∂z c + h rc ∂ϕ sv ∂z c )] = 0 for each r, s, v .(4.3.15) On the other hand, from (4 . . ∂ − ϕ ( t ))( P nr,s =1 h rs ∂ξ αj ∂z r ∂ξ βj ∂z s ) = 0 for each α, β is equivalent to n X r,s =1 ( ∂h rs ∂ ¯ z v ∂ξ αj ∂z r ∂ξ βj ∂z s + h rs ∂∂z r (cid:18) ∂ξ αj ∂ ¯ z v (cid:19) ∂ξ βj ∂z s + h rs ∂ξ αj ∂z r ∂∂z s ∂ξ βj ∂ ¯ z v ! )(4.3.16) − n X r,s,c =1 ϕ cv ( ∂h rs ∂z c ∂ξ αj ∂z r ∂ξ βj ∂z s + h rs ∂ ξ αj ∂z r ∂z c ∂ξ βj ∂z s + h rs ∂ξ αj ∂z r ∂ ξ βj ∂z s ∂z c ) = 0 for each α, β, v From (4 . . ∂ξ αj ∂ ¯ z v = P nc =1 ∂ξ αj ∂z c ϕ cv and ∂ξ βj ∂ ¯ z v = P nc =1 ∂ξ βj ∂z c ϕ cv . So (4 . .
16) is equivalent to n X r,s =1 ∂h rs ∂ ¯ z v ∂ξ αj ∂z r ∂ξ βj ∂z s + n X r,s,c =1 ( h rs ∂ ξ αj ∂z r ∂z c ϕ cv + ∂ξ αj ∂z c ∂ϕ cv ∂z r ! ∂ξ βj ∂z s + h rs ∂ξ αj ∂z r ∂ ξ βj ∂z s ∂z c ϕ cv + ∂ξ βj ∂z c ∂ϕ cv ∂z s ! )(4.3.17) − n X r,s,c =1 ϕ cv ( ∂h rs ∂z c ∂ξ αj ∂z r ∂ξ βj ∂z s + h rs ∂ ξ αj ∂z r ∂z c ∂ξ βj ∂z s + h rs ∂ξ αj ∂z r ∂ ξ βj ∂z s ∂z c )= n X r,s =1 ∂h rs ∂ ¯ z v ∂ξ αj ∂z r ∂ξ βj ∂z s + n X r,s,c =1 ( h rs ∂ξ αj ∂z c ∂ϕ cv ∂z r ∂ξ βj ∂z s + h rs ∂ξ αj ∂z r ∂ξ βj ∂z c ∂ϕ cv ∂z s ) − n X r,s,c =1 ϕ cv ∂h rs ∂z c ∂ξ αj ∂z r ∂ξ βj ∂z s = 0So (4 . .
17) is equivalent to n X r,s =1 [ ∂h rs ∂ ¯ z v + n X c =1 ( h cs ∂ϕ rv ∂z c − ϕ cv ∂h rs ∂z c + h rc ∂ϕ sv ∂z c )] ∂ξ αj ∂z r ∂ξ βj ∂z s = 0 for each α, β, v .(4.3.18)From (4 . . . .
18) is equivalent to ∂h rs ∂ ¯ z v + n X c =1 ( h cs ∂ϕ rv ∂z c − ϕ cv ∂h rs ∂z c + h rc ∂ϕ sv ∂z c ) = 0 for each r, s, v. (4.3.19)Note that (4 . .
15) is same to (4 . . . . . .
12, we note thatΛ = n X r,s =1 h rs ∂∂z r ∧ ∂∂z s = n X r,s,i,j =1 ( h rs ∂ξ αj ∂z r ∂ξ βj ∂z s ∂∂ξ αj ∧ ∂∂ξ βj + 2 h rs ∂ξ αj ∂z r ∂ ¯ ξ βj ∂z s ∂∂ξ αj ∧ ∂∂ ¯ ξ βj + h rs ∂ξ αj ∂z r ∂ ¯ ξ βj ∂z s ∂∂ ¯ ξ αj ∧ ∂∂ ¯ ξ βj )= Λ , + Λ , + Λ , . Since [Λ , Λ] = 0, the type (3 ,
0) part [Λ , Λ] , = [Λ , , Λ , ] + [Λ , , Λ , ] , = 0 (see Remark4.3.9). Since Λ , is holomorphic with respect to the complex structure induced by ϕ ( t ), we have[Λ , , Λ , ] , = 0. Hence [Λ , , Λ , ] = 0. (cid:3) Remark 4.3.20. A C ∞ complex bivector field Λ ∈ A , ( M, ∧ T M ) on M with [Λ , Λ] = 0 givesa Poisson bracket {− , −} on C ∞ complex valued functions on M . We point out that when werestrict the Poisson bracket {− , −} to holomorphic functions with respect to the complex structure M t induced by ϕ ( t ) , this is exactly the ( holomorphic ) Poisson bracket induced from Λ , when ¯ ∂ Λ − [Λ , ϕ ( t )] = 0 . Remark 4.3.21.
By Theorem . . , ϕ ( t ) in (4 . . and Λ( t ) in (4 . . satisfy ¯ ∂ Λ( t ) − [Λ( t ) , ϕ ( t )] = 0 for each t (4.3.22) and Λ( t ) , = Λ t for each t ( see Remark . . . HEOREM OF EXISTENCE AND COMPLETENESS FOR HOLOMORPHIC POISSON STRUCTURES 15
Expression of infinitesimal deformations in terms of ϕ ( t ) and Λ( t ) . In this subsection, we study how an infinitesimal deformation of ( M, Λ ) = ω − (0) in thePoisson analytic family ( M , Λ , B, ω ) (in subsection 4.1) is represented in terms of ϕ ( t ) (4.2.3)and Λ( t ) (4.3.3). Recall that an infinitesimal deformation at ( M, Λ ) is captured by an element (cid:16) ∂ ( M t , Λ t ) ∂t (cid:17) t =0 ∈ H ( M, Θ • M ) of the complex of sheaves (3.1.1) by using the following ˘Cech resolu-tion associated with the open covering U = { U j := U j × } (see Proposition 3.1.7 and Definition3.1.13). [Λ , − ] x C ( U , ∧ Θ M ) − δ −−−−→ · · · [Λ , − ] x [Λ , − ] x C ( U , ∧ Θ M ) δ −−−−→ C ( U , ∧ Θ M ) − δ −−−−→ · · · [Λ , − ] x [Λ , − ] x [Λ , − ] x C ( U , Θ M ) − δ −−−−→ C ( U , Θ M ) δ −−−−→ C ( U , Θ M ) − δ −−−−→ · · · We can also compute the hypercohomology group of the complex of sheaves (3 . .
1) by using thefollowing Dolbeault resolution. [Λ , − ] x A , ( M, ∧ T M ) ¯ ∂ −−−−→ · · · [Λ , − ] x [Λ , − ] x A , ( M, ∧ T M ) ¯ ∂ −−−−→ A , ( M, ∧ T M ) ¯ ∂ −−−−→ · · · [Λ , − ] x [Λ , − ] x [Λ , − ] x A , ( M, T M ) ¯ ∂ −−−−→ A , ( M, T M ) ¯ ∂ −−−−→ A , ( M, T M ) ¯ ∂ −−−−→ · · · We describe how a 1-cocycle in the ˘Cech resolution look like in the Dolbeault resolution. In thepicture below, we connect two resolutions. We only depict a part of resolutions that we need in thefollowing diagram. Recall that A ,p ( ∧ q T M ) is the sheaf of germs of C ∞ -section of ∧ p ¯ T ∗ M ⊗ ∧ q T M (see (1.0.1)). H ( M, ∧ Θ M ) v v ❧❧❧❧❧❧❧❧ / / C ( ∧ Θ M ) v v ❧❧❧❧❧❧❧❧ A , ( M, ∧ T M ) / / C ( A , ( ∧ T M )) H ( M, ∧ Θ M ) O O v v / / C ( ∧ Θ M ) O O v v ❧❧❧❧❧❧❧❧ δ / / C ( ∧ Θ X ) w w ♦♦♦♦♦♦♦ A , ( M, ∧ T M ) O O v v ❧❧❧❧❧❧❧❧ / / C ( A , ( ∧ T M )) O O u u ❦❦❦❦❦❦❦❦ δ / / C ( A , ( T M )) A , ( M, ∧ T M ) / / C ( A , ( ∧ T M )) C (Θ M ) v v O O − δ / / C (Θ X ) O O w w ♦♦♦♦♦♦♦ A , ( M, T M ) O O / / v v C ( A , ( T M )) [Λ0 , − ] O O − δ / / ¯ ∂ u u ❦❦❦❦❦❦❦❦ C ( A , ( T M )) ¯ ∂ v v ♠♠♠♠♠♠♠♠ O O A , ( M, T M ) O O / / C ( A , ( T M )) O O − δ / / C ( A , ( T M )) Note that each horizontal complex is exact except for edges of the “real wall”.
Now we explicitly construct the isomorphism of the first hypercohomology group from ˘Cechresolution to the first hypercohomology group from Dolbeault resolution, namely ker ( C ( U , ∧ Θ M ) ⊕ C ( U , Θ M ) → C ( U , ∧ Θ M ) ⊕ C ( U , ∧ Θ M ) ⊕ C ( U , Θ M )) im ( C ( U , Θ M ) → C ( U , ∧ Θ M ) ⊕ C ( U , Θ M ))(4.4.1) ∼ = ker ( A , ( M, ∧ T M ) ⊕ A , ( M, T M ) → A , ( M, ∧ T M ) ⊕ A , ( M, ∧ T M ) ⊕ A , ( M, T M )) im ( A , ( M, T M ) → A , ( M, ∧ T M ) ⊕ A , ( M, T M ))( b, a ) ([Λ , c ] − b, ¯ ∂c )We define the map in the following way: let ( b, a ) ∈ C ( U , ∧ Θ M ) ⊕C ( U , Θ M ) be a cohomologyclass of ˘Cech resolution. Since δa = 0, there exists a c ∈ C ( U , A , ( T M )) such that − δc = a . Since a is holomorphic ( ¯ ∂a = 0), by the commutativity ¯ ∂c ∈ A , ( M, T M ). We claim that[Λ , c ] − b ∈ A , ( M, ∧ T M ). Indeed, δ ([Λ , c ] − b ) = − [Λ , − δc ] − δb = − [Λ , a ] − δb = 0. Weshow that ( ¯ ∂c, [Λ , c ] − b ) is a cohomology class from Dolbeault resolution. Indeed, ¯ ∂ ( ¯ ∂c ) = 0 . [Λ , [Λ , c ] − b ] = 0. ¯ ∂ ([Λ , c ] − b ) + [Λ , ¯ ∂c ] = − [Λ , ¯ ∂c ] + [Λ , ¯ ∂c ] = 0. We define the map by( b, a ) ([Λ , c ] − b, ¯ ∂c ). We show that this map is well defined. Indeed, let ( b ′ , a ′ ) define the sameclass given by ( b, a ). Then there exists d ∈ C ( U , Θ M ) such that a − a ′ = − δd and b − b ′ = [Λ , d ].Let − δc ′ = a ′ . Then ¯ ∂c − ¯ ∂c ′ = ¯ ∂ ( c − c ′ − d ), and [Λ , c ] − b − ([Λ , c ′ ] − b ′ ) = [Λ , c − c ′ ] − ( b − b ′ ) =[Λ , c − c ′ − d ].For the inverse map, let ( β, α ) ∈ A , ( M, ∧ T M ) ⊕ A , ( M, T M ) be a cohomology class fromDolbeault resolution. Then there exists a c ∈ C ( U , A , ( T M )) such that ¯ ∂c = α . We define theinverse map ( β, α ) ([Λ , c ] − β, − δc ). Theorem 4.4.2. (cid:16) − (cid:16) ∂ Λ( t ) ∂t (cid:17) t =0 , (cid:16) ∂ϕ ( t ) ∂t (cid:17) t =0 (cid:17) ∈ A , ( M, ∧ T M ) ⊕ A , ( M, T M ) satisfies [Λ , − (cid:16) ∂ Λ( t ) ∂t (cid:17) t =0 ] = 0 , ¯ ∂ (cid:16) − ( ∂ Λ( t ) ∂t ) t =0 (cid:17) + [Λ , (cid:16) ∂ϕ ( t ) ∂t (cid:17) t =0 ] = 0 , ¯ ∂ (cid:16) ∂ϕ ( t ) ∂t (cid:17) t =0 = 0 , and under theisomorphism (4 . . , (cid:16) ∂ ( M t , Λ t ) ∂t (cid:17) t =0 ∈ H ( M, Θ • M ) corresponds to (cid:16) − (cid:16) ∂ Λ( t ) ∂t (cid:17) t =0 , (cid:16) ∂ϕ ( t ) ∂t (cid:17) t =0 (cid:17) Proof.
By Theorem 4.3.4 (3), (4.3.22) and (4.2.4), we have [Λ( t ) , Λ( t )] = 0, ¯ ∂ Λ( t ) − [Λ( t ) , ϕ ( t )] = 0and ¯ ∂ϕ ( t ) − [ ϕ ( t ) , ϕ ( t )] = 0 with ϕ (0) = 0 , Λ(0) = Λ . By taking the derivative of these equationswith respect to t and plugging 0, we get the first claim. Next we show the second claim. Put θ jk = n X α =1 ∂f αjk ( ξ k , t ) ∂t ! t =0 ∂∂ξ αj , σ j = n X r,s =1 (cid:18) ∂g rs ( ξ, t ) ∂t (cid:19) t =0 ∂∂ξ rj ∧ ∂∂ξ sj The infinitesimal deformation (cid:16) ∂ ( M t , Λ t ) ∂t (cid:17) t =0 ∈ H ( M, Θ • M ) is the cohomology class of the ( { σ j } , { θ jk } , ) ∈ C ( U , ∧ Θ M ) ⊕ C ( U , Θ M ) (see Proposition 3.1.7 and Definition 3.1.13). We fix a tangent vector ∂∂t ∈ T (∆), denote (cid:16) ∂f ( t ) ∂t (cid:17) t =0 by ˙ f for a C ∞ function f ( t ) , t ∈ ∆. With this notation, we put ξ j = n X α =1 ˙ ξ j α ∂∂ξ αj , where ˙ ξ j α = (cid:18) ∂ξ αj ( z, t ) ∂t (cid:19) t =0 for each j . Then we have δ { ξ j } = −{ θ jk } and ¯ ∂ξ j = n X λ =1 (cid:18) ∂ϕ λ ( z, t ) ∂t (cid:19) t =0 ∂∂z λ = n X λ =1 ˙ ϕ λ ∂∂z λ = ˙ ϕ (4.4.3)(for the detail, see [Kod05] Theorem 5.4 p.266). On the other hand, Lemma 4.4.4.
We have ˙Λ − σ j + [Λ , ξ j ] = 0 . More precisely, n X r,s =1 (cid:18) ∂h rs ( z, t ) ∂t (cid:19) t =0 ∂∂z r ∧ ∂∂z s − n X α,β =1 ∂g jαβ ( ξ j , t ) ∂t ! t =0 ∂∂ξ αj ∧ ∂∂ξ βj + [ n X r,s =1 g jrs ( ξ j , ∂∂ξ rj ∧ ∂∂ξ sj , n X c =1 ˙ ξ cj ∂∂ξ cj ] = 0 HEOREM OF EXISTENCE AND COMPLETENESS FOR HOLOMORPHIC POISSON STRUCTURES 17 equivalently ( with the notation above ) , n X r,s =1 ˙ h rs ∂∂z r ∧ ∂∂z s − n X α,β =1 ˙ g jαβ ∂∂ξ αj ∧ ∂∂ξ βj + [ n X r,s =1 g jrs ( ξ j , ∂∂ξ rj ∧ ∂∂ξ sj , n X c =1 ˙ ξ cj ∂∂ξ cj ] = 0(4.4.5) Proof.
From (4 . . . .
5) is n X r,s =1 ˙ h rs ∂∂z r ∧ ∂∂z s = n X r,s,a,b =1 ˙ h rs ∂ξ aj ( z, ∂z r ∂ξ bj ( z, ∂z s ∂∂ξ aj ∧ ∂∂ξ bj Let’s compute the third term of (4 . . n X r,s,c =1 [ g jrs ( ξ j , ∂∂ξ rj ∧ ∂∂ξ sj , ˙ ξ cj ∂∂ξ cj ] = n X r,s,c =1 ([ g jrs ( ξ j , ∂∂ξ rj , ˙ ξ cj ∂∂ξ cj ] ∧ ∂∂ξ sj − g jrs ( ξ j , ∂∂ξ sj , ˙ ξ cj ∂∂ξ cj ] ∧ ∂∂ξ rj )= n X r,s,c =1 ( g jrs ( ξ j , ∂ ˙ ξ cj ∂ξ rj ∂∂ξ cj ∧ ∂∂ξ sj − ˙ ξ cj ∂g rs ( ξ j , ∂ξ cj ∂∂ξ rj ∧ ∂∂ξ sj + g jrs ( ξ j , ∂ ˙ ξ cj ∂ξ sj ∂∂ξ rj ∧ ∂∂ξ cj )By considering the coefficients of ∂∂ξ aj ∧ ∂∂ξ bj , (4 . .
5) is equivalent to n X r,s =1 ˙ h rs ∂ξ aj ( z, ∂z r ∂ξ bj ( z, ∂z s − ˙ g jab − n X c =1 ˙ ξ cj ∂g ab ( ξ j , ∂ξ cj + n X c =1 ( g jcb ( ξ j , ∂ ˙ ξ aj ∂ξ cj + g jac ( ξ j , ∂ ˙ ξ bj ∂ξ cj ) = 0(4.4.6)On the other hand, from (4 . . g jab ( ξ j ( z, t ) , ..., ξ nj ( z, t ) , t , ..., t m ) = n X r,s =1 h rs ( z, t ) ∂ξ aj ( z, t ) ∂z r ∂ξ bj ( z, t ) ∂z s (4.4.7)By taking the derivative of (4 . .
7) with respect to t and putting t = 0, we have n X c =1 ∂g jab ( ξ j , ∂ξ cj ˙ ξ cj + ˙ g jab = n X r,s =1 ˙ h rs ∂ξ aj ( z, ∂z r ∂ξ bj ( z, ∂z s + n X r,s =1 h rs ( z, ∂ ˙ ξ aj ∂z r ∂ξ bj ( z, ∂z s + ∂ξ aj ( z, ∂z r ∂ ˙ ξ bj ∂z s )Hence (4 . .
6) is equivalent to n X c =1 g jcb ( ξ j , ∂ ˙ ξ aj ∂ξ cj + g jac ( ξ j , ∂ ˙ ξ bj ∂ξ cj = n X r,s =1 ( h rs ( z, ∂ ˙ ξ aj ∂z r ∂ξ bj ( z, ∂z s + h rs ( z, ∂ξ aj ( z, ∂z r ∂ ˙ ξ bj ∂z s )(4.4.8)Indeed, the left hand side and right hand side of (4 . .
8) coincide: from (4 . .
7) and (4 . . n X c =1 g jcb ( ξ j , ∂ ˙ ξ aj ∂ξ cj + g jac ( ξ j , ∂ ˙ ξ bj ∂ξ cj = n X r,s,c =1 ( h rs ( z, ∂ξ cj ( z, ∂z r ∂ξ bj ( z, ∂z s ∂ ˙ ξ aj ∂ξ cj + h rs ( z, ∂ξ aj ( z, ∂z r ∂ξ cj ( z, ∂z s ∂ ˙ ξ bj ∂ξ cj )= n X r,s =1 ( h rs ( z, ∂ ˙ ξ aj ∂z r ∂ξ bj ( z, ∂z s + h rs ( z, ∂ξ aj ( z, ∂z r ∂ ˙ ξ bj ∂z s )This completes Lemma 4.4.4. (cid:3) Going back to the proof of Theorem 4.4.2, we defined the isomorphism (4 . . b, a ) ([Λ , c ] − b, ¯ ∂c ) where − δc = a . We take ( b, a ) = ( { σ j } , { θ jk } ) and c = { ξ j } . Note − δ { ξ j } = { θ jk } by (4 . . . . { σ j } , { θ jk } ) is mapped to ([Λ , { ξ j } ] − { σ j } , ¯ ∂ { ξ j } ) which is( − ˙Λ , ˙ ϕ ) by Lemma 4.4.4 and (4.4.3). This completes the proof of Theorem 4.4.2. (cid:3) Integrability condition.
We have showed that given a Poisson analytic family ( M , Λ , B, ω ), the deformations ( M t , Λ t )of M = M near ( M , Λ ) is represented by the C ∞ vector (0 , ϕ ( t ) (4.2.3) and the C ∞ bivector field Λ( t ) of type (2 ,
0) (4.3.3) on M with ϕ (0) = 0 and Λ(0) = Λ satisfying the conditions:(1)[Λ( t ) , Λ( t )] = 0 , (2) ¯ ∂ Λ( t ) − [Λ( t ) , ϕ ( t )] = 0 and (3) ¯ ∂ϕ ( t ) − [ ϕ ( t ) , ϕ ( t )] = 0 for each t ∈ ∆ byTheorem 4.3.4 (3), (4.3.22) and (4.2.4).Conversely, we will show that on a compact holomorphic Poisson manifold ( M, Λ ), a C ∞ vector(0 , ϕ ∈ A , ( M, T M ) and a C ∞ type (2 ,
0) bivector field Λ ∈ A , ( M, ∧ T M ) on M such that ϕ and Λ + Λ satisfying the integrability condition (1),(2),(3) define another holomorphic Poissonstructure on the underlying differentiable manifold M . Indeed, let ϕ = P nλ =1 ϕ λv ( z ) d ¯ z v ∂∂z λ be a C ∞ vector (0 , P nr,s =1 h rs ( z ) ∂∂z r ∧ ∂∂z s be a C ∞ bivector field of type (2 ,
0) on acompact holomorphic Poisson manifold ( M, Λ ). Suppose det( δ λv − P nµ =1 ϕ µv ( z ) ϕ λµ ( z )) λ,µ =1 ,...,n = 0,and ϕ , Λ satisfy the integrability condition:[Λ + Λ , Λ + Λ] = 0(4.5.1) ¯ ∂ (Λ + Λ) − [Λ + Λ , ϕ ] = 0(4.5.2) ¯ ∂ϕ −
12 [ ϕ, ϕ ] = 0(4.5.3)Then by the Newlander-Nirenberg theorem([NN57],[Kod05]), the condition (4.5.3) gives a finiteopen covering { U j } of M and C ∞ -functions ξ αj = ξ αj ( z ) , α = 1 , ..., n on each U j such that ξ j : z → ξ j ( z ) = ( ξ j ( z ) , ..., ξ nj ( z )) gives complex coordinates on U j , and { ξ , ..., ξ j , ... } defines anothercomplex structure on M , which we denote by M ϕ . By Theorem 4.3.12, the conditions (4 . .
1) and(4 . .
2) gives a holomorphic Poisson structure (Λ + Λ) , on M ϕ . Recall that (Λ + Λ) , means thetype (2 , + Λ with respect to the complex structure induced by ϕ (see Remark 4.3.8). Remark 4.5.4.
If we replace ϕ by − ϕ , then (4 . . , (4 . . , and (4 . . are equivalent to L (Λ + ϕ ) + 12 [Λ + ϕ, Λ + ϕ ] = 0 where L = ¯ ∂ + [Λ , − ](4.5.5) which is a solution of the Maurer-Cartan equation of a differential graded Lie algebra ( g = L i ≥ g i = L p + q − i,q ≥ A ,p ( M, ∧ q T M ) , L, [ − , − ]) . This differential graded Lie algebra controls deformationsof compact holomorphic Poisson manifolds in the language of functor of Artin rings ( see the secondpart of the author’s Ph.D. thesis [Kim14]) . Example 4.5.6 (Hitchin-Goto Poisson analytic family) . Let ( M, σ ) be a compact holomorphicPoisson manifold which satisfies the ∂ ¯ ∂ -lemma. Then any class σ ([ ω ]) ∈ H ( M, Θ M ) for [ ω ] ∈ H ( M, Θ ∗ M ) is tangent to a deformation of complex structure induced by φ ( t ) = σ ( α ) where α = tω + ∂ ( t β + t β + · · · ) for (0 , -forms β i with respect to the original complex structure ( see [Hit12] Theorem 1 ) . Suppose that φ ( t ) = σ ( α ) converges for t ∈ ∆ ⊂ C . We can consider ψ = ψ ( t ) := − φ ( t ) as a C ∞ vector (0 , -form on M × ∆ , and σ as a C ∞ type (2 , bivector on M × ∆ .We note that ( ψ ( t ) , σ ) satisfies [ σ, σ ] = 0 , ¯ ∂σ − [ σ, ψ ( t )] = 0 and ∂ψ ( t ) − [ ψ ( t ) , ψ ( t )] = 0 . Thenby Newlander-Nirenberg Theorem ([NN57] , [Kod05] p. , we can give a holomorphic coordinate on M × ∆ induced by ψ . Let’s denote the complex manifold induced by ψ by M . On the other hand,the type (2 , part σ , of σ with respect to the complex structure M defines a holomorphic Poissonstructure on M . Then the natural projection π : ( M , σ , ) → ∆ defines a Poisson analytic familyof deformations of ( M, σ ) . Since σ does not depend on t , we have in the Poisson direction underthe Poisson Kodaira-Spencer map ϕ : T ∆ → H ( M, Θ • M ) by Theorem . . . More precisely, wehave ϕ ( ∂∂t ) = (0 , − σ ([ ω ])) . Theorem of existence for holomorphic Poisson structures
In this section, we prove ‘Theorem of existence for holomorphic Poisson structures’ as an analogueof ‘Theorem of existence for complex analytic structures’ by Kodaira-Spencer under the assumptionthat the associated Laplacian operator (cid:3) (induced from the operator ¯ ∂ + [Λ , − ]) is strongly ellipticand of diagonal type. HEOREM OF EXISTENCE AND COMPLETENESS FOR HOLOMORPHIC POISSON STRUCTURES 19
Statement of Theorem of existence for holomorphic Poisson structures.Theorem 5.1.1 (Theorem of existence for holomorphic Poisson structures) . Let ( M, Λ ) be acompact holomorphic Poisson manifold such that the associated Laplacian operator (cid:3) ( induced fromthe operator ¯ ∂ + [Λ , − ]) is strongly elliptic and of diagonal type. Suppose that H ( M, Θ • M ) = 0 .Then there exists a Poisson analytic family ( M , Λ , B, ω ) with ∈ B ⊂ C m satisfying the followingconditions: (1) ω − (0) = ( M, Λ )(2) The Poisson Kodaira-Spencer map ϕ : ∂∂t → (cid:16) ∂ ( M t , Λ t ) ∂t (cid:17) t =0 with ( M t , Λ t ) = ω − ( t ) is anisomorphism of T ( B ) onto H ( M, Θ • M ) : T B ϕ −→ H ( M, Θ • M ) . Let { ( π , η ) , ..., ( π m , η m ) } be a basis of H ( M, Θ • M ) where ( π λ , η λ ) ∈ A , ( M, ∧ T M ) ⊕ A , ( M, T M )for λ = 1 , ..., m . Let ∆ ǫ = { t ∈ C m || t | < ǫ } for some ǫ >
0. Assume that there is a family { ( ϕ ( t ) , Λ( t )) | t ∈ ∆ ǫ } of C ∞ vector (0 , ϕ ( t ) = P nλ =1 P nv =1 ϕ λv ( z, t ) d ¯ z v ∂∂z λ ∈ A , ( M, T M )and C ∞ type (2 ,
0) bivectors Λ( t ) = P nα,β =1 Λ αβ ( z, t ) ∂∂z α ∧ ∂∂z β ∈ A , ( M, ∧ T M ) on M , whichsatisfy(1) [Λ( t ) , Λ( t )] = 0(2) ¯ ∂ Λ( t ) − [Λ( t ) , ϕ ( t )] = 0(3) ¯ ∂ϕ ( t ) − [ ϕ ( t ) , ϕ ( t )] = 0and the initial conditions ϕ (0) = 0 , Λ(0) = Λ , ( − (cid:18) ∂ Λ( t ) ∂t λ (cid:19) t =0 , (cid:18) ∂ϕ ( t ) ∂t λ (cid:19) t =0 ) = ( − π λ , − η λ ) , λ = 1 , ..., m, Since ϕ (0) = 0, we may assume det( δ λv − P nµ =1 ϕ µv ( z, t ) ϕ λµ ( z, t )) λ,µ =1 ,...,n = 0 if ∆ ǫ is sufficientlysmall. Therefore, as in subsection 4.5, by the Newlander-Nirenberg theorem([NN57],[Kod05] p.268),each ϕ ( t ) determines a complex structure M ϕ ( t ) on M . The conditions (2) and (3) imply that(2 , t ) , of Λ( t ) with respect to the complex structure induced from ϕ ( t ) is a holomorphicPoisson structure on M ϕ ( t ) . If the family { ( M ϕ ( t ) , Λ( t ) , ) | t ∈ ∆ ǫ } is a Poisson analytic family,it satisfies the conditions (1) and (2) in Theorem 5.1.1 by Theorem 4.4.2. We will constructsuch a family { ( ϕ ( t ) , Λ( t )) | t ∈ ∆ ǫ } under the assumption H ( M, Θ • M ) = 0 and then show that { ( M ϕ ( t ) , Λ( t ) , ) | t ∈ ∆ ǫ } is a Poisson analytic family in the subsection 5.3, which completes theproof of Theorem 5 . . Remark 5.1.2.
By replacing ϕ ( t ) by − ϕ ( t ) , it is sufficient to construct ϕ ( t ) and Λ( t ) satisfying (1) [Λ( t ) , Λ( t )] = 0(2) ¯ ∂ Λ( t ) + [Λ( t ) , ϕ ( t )] = 0(3) ¯ ∂ϕ ( t ) + [ ϕ ( t ) , ϕ ( t )] = 0 and the initial conditions ϕ (0) = 0 , Λ(0) = Λ , ( (cid:18) ∂ Λ( t ) ∂t λ (cid:19) t =0 , (cid:18) ∂ϕ ( t ) ∂t λ (cid:19) t =0 ) = ( π λ , η λ ) , λ = 1 , ..., m, (5.1.3) We note that (1) , (2) , (3) are equivalent to (5.1.4) ¯ ∂ ( ϕ ( t ) + Λ( t )) + 12 [ ϕ ( t ) + Λ( t ) , ϕ ( t ) + Λ( t )] = 0We construct such α ( t ) := ϕ ( t ) + Λ( t ) in the following subsection.5.2. Construction of α ( t ) = ϕ ( t ) + Λ( t ) . We use the Kuranishi’s method presented in [MK06] to construct α ( t ). First we note the follow-ing: let A p = A ,p ( M, T M ) ⊕ · · · ⊕ A , ( M, ∧ p +1 T M ) and L = ¯ ∂ + [Λ , − ]. Then the sequence A L −→ A L −→ A L −→ · · · is an elliptic complex. So we have the adjoint operator L ∗ , Green’s operator G , Laplacian operator (cid:3) := LL ∗ + L ∗ L and H where H is the orthogonal projection to the (cid:3) -harmonic subspace H of L p ≥ A p . In particular we have H : A p → H p ∼ = H p ( M, Θ • M ). For the detail, we refer to [Wel08].We introduce the H¨older norms in the spaces A p = A ,p ( M, T M ) ⊕ · · · ⊕ A , ( M, ∧ p +1 T M ) inthe following way: we fix a finite open covering { U j } of M such that z j = ( z j , ..., z nj ) are localcoordinates on U j . Let φ ∈ A p which is locally expressed on U j as φ = X r + s = p +1 ,s ≥ φ jα ··· α r β ··· β s ( z ) d ¯ z α j ∧ · · · ∧ d ¯ z α r j ∧ ∂∂z β j ∧ · · · ∧ ∂∂z β s j Let k ∈ Z , k ≥ , θ ∈ R , < θ <
1. Let h = ( h , ..., h n ) , h i ≥ , | h | := P ni =1 h i where n = dim M .Then denote D hj = ∂∂x j ! h · · · ∂∂x nj ! h n , z αj = x α − j + ix αj Then the H¨older norm || ϕ || k + θ is defined as follows: || φ || k + θ = max j { X h, | h |≤ k sup z ∈ U j | D hj φ jα ··· α r β ··· β s ( z ) | ! + sup y,z ∈ U j , | h | = k | D hj φ jα ··· α r β ··· β s ( y ) − D hj φ jα ··· α r β ··· β s ( z ) || y − z | θ } , where the sup is over all α , ..., α r , β , ..., β s .Now suppose that the associated Laplacian operator (cid:3) induced from L = ¯ ∂ + [Λ , − ] is a stronglyelliptic operator whose principal part is of diagonal type. Then by [Kod05] Appendix Theorem 4.3page 436, we have a priori estimate(5.2.1) || φ || k + θ ≤ C ( || (cid:3) φ || k − θ + || φ || )where k ≥ C is a constant which is independent of ϕ and || φ || = max j,α ,...,β s sup z ∈ U j | φ jα ··· α r β ··· β s ( z ) | .We will use the following two lemmas. Lemma 5.2.2.
For φ, ψ ∈ A , we have || [ φ, ψ ] || k + θ ≤ C || φ || k +1+ θ || ψ || k +1+ θ , where C is indepen-dent of φ and ψ . Lemma 5.2.3.
For φ ∈ A , we have || Gφ || k + θ ≤ C || φ || k − θ , k ≥ , where C depends only on k and θ , not on φ .Proof. This follows from (5.2.1). See [MK06] p.160 Proposition 2.3. (cid:3)
With this preparation, we construct α ( t ) := ϕ ( t ) + Λ( t ) = Λ + P ∞ µ =1 ( ϕ µ ( t ) + Λ µ ( t )), where ϕ µ ( t ) + Λ µ ( t ) = X v + ··· + v m = µ ( ϕ v ··· v m + Λ v ··· v m ) t v · · · t v m m with ϕ v ··· v m + Λ v ··· v m ∈ A , ( M, T M ) ⊕ A , ( M, ∧ T M ) such that¯ ∂α ( t ) + 12 [ α ( t ) , α ( t )] = 0 , α ( t ) = ϕ ( t ) + Λ ( t ) = m X v =1 ( η v + π v ) t v , (5.2.4)where { η v + π v } is a basis for H ∼ = H ( M, Θ • M ) (see (5.1.3)). Let β ( t ) := α ( t ) − Λ = P ∞ µ =1 ( ϕ µ ( t ) +Λ µ ( t )). Then (5 . .
4) is equivalent to Lβ ( t ) + 12 [ β ( t ) , β ( t )] = 0 , β ( t ) = α ( t )(5.2.5)Constructing α ( t ) is equivalent to constructing β ( t ). We will construct β ( t ) satisfying (5 . . β ( t ) = β ( t ) − L ∗ G [ β ( t ) , β ( t )] , where β ( t ) = α ( t ). Then (5 . .
6) has a unique formal power series solution β ( t ) = P ∞ µ =1 β µ ( t ),and there exists a ǫ > t ∈ ∆ ǫ = { t ∈ C m || t | < ǫ } , β ( t ) = P ∞ µ =1 β µ ( t ) converges in HEOREM OF EXISTENCE AND COMPLETENESS FOR HOLOMORPHIC POISSON STRUCTURES 21 the norm || · || k + θ (for the detail, see [MK06] p.162 Proposition 2.4. By virtue of the integrabilitycondition (5.2.5), we can formally apply their argument.). Proposition 5.2.7. β ( t ) satisfies Lβ ( t ) + [ β ( t ) , β ( t )] = 0 if and only if H [ β ( t ) , β ( t )] = 0 , where H : A = A , ( M, T M ) ⊕ A , ( M, ∧ T M ) ⊕ A , ( M, ∧ T M ) → H ∼ = H ( M, Θ • M ) is the orthogonalprojection to the harmonic subspace of A .Proof. We simply note that ( L i ≥ g i , g i = L p + q − i,p ≥ ,q ≥ A ,p ( M, ∧ q T M ) , L = ¯ ∂ +[Λ , − ] , [ − , − ])is a differential graded Lie algebra and so the argument in the proof of [MK06] p.163 Proposition2.5 is formally applied to our case by Lemma 5.2.2 and Lemma 5.2.3. (cid:3) Now suppose that H ( M, Θ • M ) = 0. Then by Proposition 5.2.7, β ( t ) satisfies (5 . .
5) for t ∈ ∆ ǫ .Hence α ( t ) = β ( t ) + Λ = ϕ ( t ) + Λ( t ) is the desired one satisfying (5 . . α ( t ) hasthe following property which we need in the construction of a Poisson analytic family in the nextsubsection. Proposition 5.2.8. α ( t ) = β ( t ) + Λ = ϕ ( t ) + Λ( t ) is C ∞ in ( z, t ) and holomorphic in t .Proof. We note that (cid:3) is a strongly elliptic differential operator whose principal part is of diagonaltype by our assumption. So we can formally apply the argument of [MK06] p.163 Proposition 2.6.See also [Kod05] Appendix p.452 § (cid:3) Construction of a Poisson analytic family.
In the previous subsection, we have constructed a family { ( ϕ ( t ) , Λ( t )) | t ∈ ∆ ǫ } of C ∞ vector(0 , ϕ ( t ) and C ∞ type (2 ,
0) bivectors Λ( t ) ϕ ( t ) = n X λ =1 n X v =1 ϕ λv ( z, t ) d ¯ z v ∂∂z λ , Λ( t ) = n X α,β =1 Λ αβ ( z, t ) ∂∂z α ∧ ∂∂z β satisfying the integrability condition [Λ( t ) , Λ( t )] = 0 , ¯ ∂ Λ( t ) = [Λ( t ) , ϕ ( t )] , ¯ ∂ϕ ( t ) = [ ϕ ( t ) , ϕ ( t )] andthe initial conditions ϕ (0) = 0 , Λ(0) = Λ , ( − (cid:16) ∂ Λ( t ) ∂t λ (cid:17) t =0 , (cid:16) ∂ϕ ( t ) ∂t λ (cid:17) t =0 ) = ( − π λ , − η λ ) , λ = 1 , ..., m ,where ϕ λv ( z, t ) and Λ αβ ( z, t ) are C ∞ functions of z , ..., z n , t , ..., t m and holomorphic in t , ..., t m .( ϕ ( t ) , Λ( t )) determines a holomorphic Poisson structure ( M ϕ ( t ) , Λ( t ) , ) on M for each t ∈ ∆ ǫ . Inorder to show that { ( M ϕ ( t ) , Λ( t ) , ) | t ∈ ∆ ǫ } is a Poisson analytic family, we consider ϕ := ϕ ( t ) as avector (0 , M × ∆ ǫ , and Λ := Λ( t ) as a (2 ,
0) bivector on M × ∆ ǫ .Then since ϕ λv = ϕ λv ( z, t ) are holomorphic in t , ..., t m (Proposition 5.2.8), we have ∂ϕ λv ∂ ¯ t µ = 0 in¯ ∂ϕ = n X λ,v =1 n X β =1 ∂ϕ λv ∂ ¯ z β d ¯ z β + m X µ =1 ∂ϕ λv ∂ ¯ t µ d ¯ t µ ∧ d ¯ z v ∂∂z λ Similarly since Λ αβ ( z, t ) is holomorphic in t , ..., t m (Proposition 5.2.8), we have ∂ Λ αβ ∂ ¯ t µ = 0 in¯ ∂ Λ = X α,β n X v =1 ∂ Λ αβ ∂ ¯ z v d ¯ z v + m X µ =1 ∂ Λ αβ ∂ ¯ t µ d ¯ t µ ∂∂z α ∧ ∂∂z β Hence ϕ and Λ satisfy ¯ ∂ϕ = [ ϕ, ϕ ], ¯ ∂ Λ = [Λ , ϕ ], and [Λ , Λ] = 0. Then by the Newlander-Nirenbergtheorem([NN57],[Kod05] p.268), ϕ defines a complex structure M on M × ∆ ǫ and (2 , , of Λ defines a holomorphic Poisson structure on M . Let ω : M → ∆ ǫ be the natural projection.Then { ( M ϕ ( t ) , Λ( t ) , ) | t ∈ ∆ ǫ } forms a Poisson analytic family ( M , Λ , , ∆ ǫ , ω ) (for the detail, see[Kod05] p.282). This completes the proof of Theorem 5 . . Theorem of completeness for holomorphic Poisson structures
Statement of Theorem of completeness for holomorphic Poisson structures.
Change of parameters. (compare [Kod05] p.205)Consider a Poisson analytic family { ( M t , Λ t ) | ( M t , Λ t ) = ω − ( t ) , t ∈ B } = ( M , Λ , B, ω ) of com-pact holomorphic Poisson manifolds, where B is a domain of C m . Let D be a domain of C r and h : s → t = h ( s ) , s ∈ D , a holomorphic map of D into B . Then by changing the parameter from t to s , we will construct a Poisson analytic family { ( M h ( t ) , Λ h ( t ) ) | s ∈ D } on the parameter space D in the following.Let M× B D := { ( p, s ) ∈ M× B | ω ( p ) = h ( s ) } . Then we have the following commutative diagram M × B D p −−−−→ M π y y ω D h −−−−→ B such that ( M× B D, D, π ) is a complex analytic family in the sense of Kodaira-Spencer and π − ( s ) = M h ( s ) . We show that ( M × B D, D, π ) is naturally a Poisson analytic family such that π − ( s ) =( M h ( s ) , Λ h ( s ) ) and p is a Poisson map. Note that the bivector field Λ on M can be considered as abivector field on M × D which gives a holomorphic Poisson structure on M × D . So ( M × D, Λ)is a holomorphic Poisson manifold. We show that
M × B D is a holomorphic Poisson submanifoldof ( M × D, Λ) and defines a Poisson analytic family. Let ( p , s ) ∈ M × B D . Taking a sufficientlysmall coordinate polydisk ∆ with h ( s ) ∈ ∆, we represent ( M ∆ , Λ ∆ ) = ω − (∆) in the form of( M ∆ , Λ ∆ ) = ( l [ j =1 U j × ∆ , n X α,β =1 g jαβ ( z j , t ) ∂∂z αj ∧ ∂∂z βj )where each U j is a polydisk independent of t , and ( z j , t ) ∈ U j × ∆ and ( z k , t ) ∈ U k × ∆ are thesame point on M ∆ if z αj = f αjk ( z k , t ) , α = 1 , ..., n . Let E be a sufficiently small polydisk of D suchthat s ∈ E and h ( E ) ⊂ ∆. Then we can represent ( M × D, Λ) around ( p , s ) in the form of( M ∆ × E, Λ | M ∆ × E ) = ( l [ j =1 U j × ∆ × E, n X α,β =1 g jαβ ( z j , t ) ∂∂z αj ∧ ∂∂z βj )where ( z j , t, s ) ∈ U j × ∆ × E and ( z k , t, s ) ∈ U k × ∆ × E are the same point on M ∆ × E if z j = f jk ( z k , t ). Then we can represent M × B D around ( p , s ) in the form of S lj =1 U j × G E ,where G E = { ( h ( s ) , s ) | s ∈ E } ⊂ ∆ × E , and ( z j , h ( s ) , s ) ∈ U j × G E and ( z k , h ( s ) , s ) ∈ U k × G E are the same point if z j = f jk ( z k , h ( s )). We note that at ( p , s ) ∈ M × B D ⊂ M × D , we haveΛ ( p ,s ) = P nα,β =1 g jαβ ( p , h ( s )) ∂∂z αj | p ∧ ∂∂z βj | p ∈ ∧ T M× B D . Hence M × B D is a holomorphicPoisson submanifold of ( M × D, Λ), and p : ( M × B D, Λ | M× B D ) → ( M , Λ) is a Poisson map.Since G E is biholomorphic to E . The holomorphic Poisson manifold ( M × B D, Λ | M× B D ) isrepresented locally by the form( l [ j =1 U j × E, n X α,β =1 g jαβ ( z j , h ( s )) ∂∂z αj ∧ ∂∂z βj )where ( z k , s ) ∈ U k × E and ( z j , s ) ∈ U j × E are the same point if z j = f jk ( z k , h ( s )), which showsthat ( M × B D, D, Λ | M× B D , π ) is a Poisson analytic family and π − ( s ) = ( M h ( s ) , Λ h ( s ) ). Definition 6.1.1.
The Poisson analytic family ( M × B D, D, Λ | M× B D , π ) is called the Poissonanalytic family induced from ( M , B, Λ , ω ) by the holomorphic map h : D → B . We point out that change of variable formula holds for infinitesimal Poisson deformations as ininfinitesimal deformations of complex structures ([Kod05] Theorem 4.7 p.207).
Theorem 6.1.2.
For any tangent vector ∂∂s = c ∂∂s + · · · + c r ∂∂s r ∈ T s ( D ) , the infinitesimal Poissondeformation of ( M h ( s ) , Λ h ( s ) ) along ∂∂s is given by ∂ ( M h ( s ) , Λ h ( s ) ) ∂s = ( m X λ =1 ∂t λ ∂s ∂M t ∂t λ , m X λ =1 ∂t λ ∂s ∂ Λ t ∂t λ )With this preparation, we discuss a concept of completeness and ‘Theorem of completeness’ inthe context of deformations of compact holomorphic Poisson manifolds in the next subsection. HEOREM OF EXISTENCE AND COMPLETENESS FOR HOLOMORPHIC POISSON STRUCTURES 23
Statement of ‘Theorem of completeness for holomorphic Poisson structures’.
Definition 6.1.3.
Let ( M , Λ M , B, ω ) be a Poisson analytic family of compact holomorphic Poissonmanifolds, and t ∈ B . Then ( M , Λ M , B, ω ) is called complete at t ∈ B if for any Poisson analyticfamily ( N , Λ N , D, π ) such that D is a domain of C l containing and that π − (0) = ω − ( t ) , thereis a sufficiently small domain ∆ with ∈ ∆ ⊂ D , and a holomorphic map h : s → t = h ( s ) with h (0) = t such that ( N ∆ , Λ N ∆ , ∆ , π ) is the Poisson analytic family induced from ( M , Λ M , B, ω ) by h where ( N ∆ , Λ N ∆ , ∆ , π ) is the restriction of ( N , Λ N , D, π ) to ∆ ( see Remark . . . We will prove the following theorem which is an analogue of ‘Theorem of completeness’ byKodaira-Spencer (see Theorem 1.0.4).
Theorem 6.1.4 (Theorem of completeness for holomorphic Poisson structures) . Let ( M , Λ M , B, ω ) be a Poisson analytic family of deformations of a compact holomorphic Poisson manifold ( M , Λ ) = ω − (0) , B a domain of C m containing . If the Poisson Kodaira-Spencer map ϕ : T ( B ) → H ( M , Θ • M ) is surjective, the Poisson analytic family ( M , Λ M , B, ω ) is complete at ∈ B . Remark 6.1.5.
In order to prove Theorem . . , as in [Kod05] Lemma . p. , it suffices toshow that for any given Poisson analytic family ( N , Λ N , D, π ) with π − (0) = ( M , Λ ) , if we takea sufficiently small domain ∆ with ∈ ∆ ⊂ D , we can construct a holomorphic map h : s → t = h ( s ) , h (0) = 0 , of ∆ into B , and a Poisson holomorphic map g of ( N ∆ , Λ N ∆ ) = π − (∆) into ( M , Λ M ) satisfying the following condition: g is a Poisson holomorphic map extending the identity g : π − (0) = ( M , Λ ) → ( M , Λ ) , and g maps each ( N s , Λ N s ) = π − ( s ) Poisson biholomorphicallyonto ( M h ( s ) , Λ M h ( s ) ) . We will construct such h and g by extending Kodaira’s elementary method ( see [Kod05] Chapter . Preliminaries.
We extend the argument of [Kod05] p.285-286 (to which we refer for the detail) in the contextof a Poisson analytic family. We tried to keep notational consistency with [Kod05].Since the problem is local with respect to B , we may assume that B = { t ∈ C m || t | < } , and( M , Λ M , B, ω ) is written in the following form( M , Λ M ) = [ j ( U j , Λ M j ) , U j = { ( ξ j , t ) ∈ C n × B || ξ j | < } where the Poisson structure Λ M is given by Λ M j = P nr,s =1 Λ r,sM j ( ξ j , t ) ∂∂ξ rj ∧ ∂∂ξ sj on U j with Λ r,sM j ( ξ j , t ) = − Λ s,rM j ( ξ j , t ), and ω ( ξ j , t ) = t . For U j ∩ U k = ∅ , ( ξ j , t ) and ( ξ k , t ) are the same point of M if ξ j = g jk ( ξ k , t ) = ( g jk ( ξ k , t ) , ..., g njk ( ξ k , t )) , (6.2.1)where g αjk ( ξ k , t ), α = 1 , ..., n , are holomorphic functions on U j ∩ U k , and we have the followingrelations Λ r,sM j ( g jk ( ξ k , t ) , t ) = n X p,q =1 Λ p,qM k ( ξ k , t ) ∂g rjk ∂ξ pk ∂g sjk ∂ξ qk . (6.2.2)Similarly we assume that D = { s ∈ C l || s | < } , and ( N , Λ N , D, π ) is written in the followingform ( N , Λ N ) = [ j ( W j , Λ N j ) , W j = { ( z j , s ) ∈ C n × D || z j | < } where the Poisson structure Λ N is given by Λ N j = P nα,β =1 Λ α,βN j ( z j , t ) ∂∂z αj ∧ ∂∂z βj on W j withΛ α,βN j ( z j , t ) = − Λ β,αN j ( z j , t ), and π ( z j , s ) = s . For W j ∩ W k = ∅ , ( z j , s ) and ( z k , s ) are the samepoint of N if z j = f jk ( z k , s ) = ( f jk ( z k , s ) , ..., f njk ( z k , s )) . (6.2.3)and we have Λ α,βN j ( f jk ( z k , s ) , s ) = n X a,b =1 Λ a,bN k ( z k , s ) ∂f αjk ∂z ak ∂f βjk ∂z bk (6.2.4) Since ( N , Λ N ) = π − (0) = ( M , Λ ) = ω − (0) = ( M , Λ M ), we may assume ( W j ∩ N , Λ N j ) =( U j ∩ M , Λ M j ) where Λ N j := P nα,β =1 Λ α,βN j ( z j , ∂∂z αj ∧ ∂∂z βj , and Λ M j := P nr,s =1 Λ r,sM j ( ξ j , ∂∂ξ rj ∧ ∂∂ξ sj ,and assume that the local coordinates ( ξ j ,
0) and ( z j ,
0) coincide on W j ∩ N = U j ∩ M . In otherwords, if ξ j = z j , ..., ξ nj = z nj , ( ξ j ,
0) and ( z j ,
0) are the same point of W j ∩ N = U j ∩ M , and wehave Λ α,βN j ( z j ,
0) = Λ α,βM j ( ξ j , b jk ( ξ k ) := g jk ( ξ k , , Λ α,βM j ( ξ j ) := Λ α,βM j ( ξ j , . .
1) and (6 . . b jk ( z k ) = f jk ( z k , , Λ α,βM j ( z j ) = Λ α,βM j ( z j ,
0) = Λ α,βN j ( z j , N , Λ N ) = ( M , Λ M = Λ ) = [ j ( U j , Λ M j ) , U j = W j ∩ N = U j ∩ M , (6.2.7)such that { z j } , z j = ( z j , ..., z nj ), is a system of local complex coordinates of the complex manifold N = M with respect to { U j } , and the Poisson structure is given by Λ M j = P nα,β =1 Λ α,βM j ( z j ) ∂∂z αj ∧ ∂∂z βj on U j with Λ α,βM j ( z j ) = − Λ β,αM j ( z j ). The coordinate transformation on U j ∩ U k is given by z αj = b αjk ( z k ) , α = 1 , ..., n, and we haveΛ α,βM j ( b jk ( z k )) = n X a,b =1 Λ a,bM k ( z k ) ∂b αjk ∂z ak ∂b βjk ∂z bk . (6.2.8)6.3. Construction of Formal Power Series.
As in Remark 6.1.5, we have to define a holomorphic map h : s → t = h ( s ) with h (0) = 0of ∆ = { s ∈ D || s | < ǫ } into B for a sufficiently small ǫ >
0, and to extend the identity g :( N , Λ ) → ( M = N , Λ ) to a Poisson holomorphic map g : π − (∆) = ( N ∆ , Λ N ∆ ) → ( M , Λ) suchthat ω ◦ g = h ◦ π .We begin with constructing formal power series h ( s ) = P ∞ v =1 h v ( s ) of s , ..., s l where h v ( s )is a homogenous polynomial of degree v of s , ..., s l , and formal power series g j ( z j , s ) = z j + P ∞ v =1 g j | v ( z j , s ) in terms of s , ..., s l for each U j in (6.2.7), whose coefficients are vector valuedholomorphic functions on U j where g j | v ( z j , s ) = P v + ··· + v l = v g jv ··· v l ( z j ) s v · · · s v l l is a homogeneouspolynomial of degree v of s , ..., s l , and each component g αjv ··· v n ( z j ) , α = 1 , ..., n of the coefficient g jv ...v l ( z j ) = ( g jv ··· v l ( z j ) , ..., g njv ··· v l ( z j )) is a holomorphic function of z j , ..., z nj defined on U j . Theformal power series h ( s ) and g j ( z j , s ) will satisfy g j ( f jk ( z k , s ) , s ) = g jk ( g k ( z k , s ) , h ( s )) on U j ∩ U k = ∅ (6.3.1) Λ r,sM j ( g j ( z j , s ) , h ( s )) = n X α,β =1 Λ α,βN j ( z j , s ) ∂g rj ∂z αj ∂g sj ∂z βj on U j . (6.3.2)For the meaning of (6.3.1), we refer to [Kod05] p.286-288. (6.3.1) is a crucial condition for theproof of ‘Theorem of completeness for complex analytic structures’ (Theorem 1.0.4). However, inorder to prove ‘Theorem of completeness for holomorphic Poisson structures’ (Theorem 6.1.4), weneed to impose additional condition (6 . .
2) which means that g j ( z j , s ) is a Poisson map.We will write h v ( s ) := h ( s ) + · · · + h v ( s ) .g vj ( z j , s ) := z j + g j | ( z j , s ) + · · · g j | v ( z j , s ) . The equalities (6.3.1) and (6.3.2) are equivalent to the following system of the infinitely manycongruence: g vj ( f jk ( z k , s ) , s ) ≡ v g jk ( g vk ( z k , s ) , h v ( s ))(6.3.3) Λ r,sM j ( g vj ( z j , s ) , h v ( s )) ≡ v n X α,β =1 Λ α,βN j ( z j , s ) ∂g rj v ∂z αj ∂g sj v ∂z βj (6.3.4) HEOREM OF EXISTENCE AND COMPLETENESS FOR HOLOMORPHIC POISSON STRUCTURES 25 for v = 0 , , , , ... where we indicate by ≡ v that the power series expansions with respect to s ofboth sides of (6.3.3) and (6.3.4) coincide up to the term of degree v .We will construct h v ( s ) , g vj ( z j , s ) satisfying (6 . . v and (6 . . v inductively on v . Then theresulting formal power series h ( s ) and g j ( z j , s ) will satisfy (6 . .
1) and (6 . . v = 0, since h ( s ) = 0 and g j ( z j , s ) = z j , (6 . . and (6 . . hold by (6 . . , (6 . . h v − ( s )and g v − j ( z j , s ) are already constructed in such a manner that, for each U j ∩ U k = ∅ , g v − j ( f jk ( z k , s ) , s ) ≡ v − g jk ( g v − k ( z k , s ) , h v − ( s ))(6.3.5)and for each U j , Λ r,sM j ( g v − j ( z j , s ) , h v − ( s )) ≡ v − n X α,β =1 Λ α,βN j ( z j , s ) ∂g rj v − ∂z αj ∂g sj v − ∂z βj (6.3.6)hold. We will find h v ( s ) and g j | v ( z j , s ) such that h v ( s ) = h v − ( s ) + h v ( s ), and g vj ( z j , s ) = g v − j ( z j , s ) + g j | v ( z j , s ) satisfy (6 . . v on each U j ∩ U k and (6 . . v on each U j .For this purpose, we start from finding the equivalent conditions to (6 . . v and (6 . . v , andthen interpret them cohomologically by using ˘Cech resolution of the complex of sheaves (3.1.1)with respect to the open covering (6 . .
7) of M = N (see Lemma 6.3.22 below).For the equivalent condition to (6 . . v , we briefly summarize Kodaira’s result in the following: ifwe let Γ jk | v denote the sum of the terms of degree v of g v − j ( f jk ( z k , s ) , s ) − g jk ( g v − k ( z k , s ) , h v − ( s )):Γ jk ( z j , s ) ≡ v g v − j ( f jk ( z k , s ) , s ) − g jk ( g v − k ( z k , s ) , h v − ( s )) , (6.3.7)then (6 . . v is equivalent to the following:Γ jk | v ( z j , s ) = n X β =1 ∂z j ∂z βk g βk | v ( z k , s ) − g j | v ( z j , s ) + m X u =1 (cid:18) ∂g jk ( z k , t ) ∂t u (cid:19) t =0 h u | v ( s )(6.3.8)where z k and z j = b jk ( z k ) are the local coordinates of the same point of N = M (for the detail,see [Kod05] p.289-290).On the other hand, let’s find the equivalent condition to (6 . . v . We note thatΛ r,sM j ( g vj ( z j , s ) , h v ( s )) = Λ r,sM j ( g v − j ( z j , s ) + g j | v ( z j , s ) , h v − ( s ) + h v ( s ))(6.3.9)By expanding Λ r,sM j ( ξ j + ξ, t + ω ) into power series of ξ , ..., ξ n , ω , ..., ω m , we obtainΛ r,sM j ( ξ j + ξ, t + ω ) = Λ r,sM j ( ξ j , t ) + n X β =1 ∂ Λ r,sM j ∂ξ βj ( ξ j , t ) ξ β + m X u =1 ∂ Λ r,sM j ∂t u ( ξ j , t ) ω u + · · · (6.3.10)where · · · denotes the terms of degree ≥ ξ , ..., ξ n , ω , ..., ω m . Let’s consider the left hand sideof (6 . . v . Then from (6 . . . . . . r,sM j ( g vj ( z j , s ) , h v ( s )) − Λ r,sM j ( g v − j ( z j , s ) , h v − ( s ))(6.3.11) ≡ v n X β =1 ∂ Λ r,sM j ∂ξ βj ( g v − j ( z j , s ) , h v − ( s )) g βj | v ( z j , s ) + m X u =1 ∂ Λ r,sM j ∂t u ( g v − j ( z j , s ) , h v − ( s )) h u | v ( s ) ≡ v n X β =1 ∂ Λ r,sM j ∂ξ βj ( g v − j ( z j , , h v − (0)) g βj | v ( z j , s ) + m X u =1 ∂ Λ r,sM j ∂t u ( g v − j ( z j , , h v − (0)) h u | v ( s )= n X β =1 ∂ Λ r,sM j ∂z βj g βj | v ( z j , s ) + m X u =1 ∂ Λ r,sM j ( z j , t ) ∂t u ! t =0 h u | v ( s ) On the other hand, let’s consider the right hand side of (6 . . v . Then from (6 . . n X α,β =1 Λ α,βN j ( z j , s ) ∂g rj v ∂z αj ∂g sj v ∂z βj = n X α,β =1 Λ α,βN j ( z j , s ) ∂ ( g rj v − + g rj | v ) ∂z αj ∂ ( g sj v − + g sj | v ) ∂z βj (6.3.12) ≡ v n X α,β =1 Λ α,βN j ( z j , s ) ∂g rj v − ∂z αj ∂g sj v − ∂z βj + n X α,β =1 Λ α,βN j ( z j , s ) ∂g rj v − ∂z αj ∂g sj | v ∂z βj + n X α,β =1 Λ α,βN j ( z j , s ) ∂g rj | v ∂z αj ∂g sj v − ∂z βj ≡ v n X α,β =1 Λ α,βN j ( z j , s ) ∂g rj v − ∂z αj ∂g sj v − ∂z βj + n X β =1 Λ r,βM j ( z j ) ∂g sj | v ∂z βj + n X α =1 Λ α,sM j ( z j ) ∂g rj | v ∂z αj Then from (6 . .
11) and (6 . . . . v is equivalent to the following: − Λ r,sM j ( g v − j ( z j , s ) , h v − ( s )) + n X α,β =1 Λ α,βN j ( z j , s ) ∂g rj v − ∂z αj ∂g sj v − ∂z βj (6.3.13) ≡ v n X β =1 ∂ Λ r,sM j ∂z βj g βj | v ( z j , s ) + m X u =1 ∂ Λ r,sM j ( z j , t ) ∂t u ! t =0 h u | v ( s ) − n X β =1 Λ r,βM j ( z j ) ∂g sj | v ∂z βj − n X α =1 Λ α,sM j ( z j ) ∂g rj | v ∂z αj By induction hypothesis (6.3.6), the left hand side of (6.3.13) ≡ v −
0. Hence if we let λ r,sj | v denotethe terms of degree v of the left hand side of (6.3.13), we have λ r,sj | v ( z j , s ) ≡ v − Λ r,sM j ( g v − j ( z j , s ) , h v − ( s )) + n X α,β =1 Λ α,βN j ( z j , s ) ∂g rj v − ∂z αj ∂g sj v − ∂z βj (6.3.14)Hence from (6 . .
13) and (6 . . . . v is equivalent to the following: λ r,sj | v ( z j , s ) = n X β =1 ∂ Λ r,sM j ∂z βj g βj | v ( z j , s ) + m X u =1 ∂ Λ r,sM j ( z j , t ) ∂t u ! t =0 h u | v ( s ) − n X β =1 Λ r,βM j ( z j ) ∂g sj | v ∂z βj − n X α =1 Λ α,sM j ( z j ) ∂g rj | v ∂z αj (6.3.15)where z k and z j = b jk ( z k ) are the local coordinates of the same point of N = M . We note that λ r,sj | v ( z j , s ) = − λ s,rj | v ( z j , s ).As in [Kod05] p.291, to interpret the meaning of (6 . . v , and (6 . . v in terms of ˘Cech resolutionof the complex of sheaves (3 . .
1) with respect to the open covering (6 . .
7) of M = N , we introduceholomorphic vector fields and bivector fields as follows: θ ujk = n X α =1 θ αujk ( z j ) ∂∂z αj = n X α =1 ∂g αjk ( z k , t ) ∂t u ! t =0 ∂∂z αj , z k = b jk ( z j )(6.3.16) Λ ′ uj = n X r,s =1 Λ ′ r,suj ( z j ) ∂∂z rj ∧ ∂∂z sj := n X r,s =1 ∂ Λ r,sM j ( z j , t ) ∂t u ! t =0 ∂∂z rj ∧ ∂∂z sj (6.3.17) Γ jk | v ( s ) = n X α =1 Γ αjk | v ( z j , s ) ∂∂z αj (6.3.18) g k | v ( s ) = n X β =1 g βk | v ( z k , s ) ∂∂z βk (6.3.19) λ j | v ( s ) = n X r,s =1 λ r,sj | v ( z j , s ) ∂∂z rj ∧ ∂∂z sj (6.3.20)By (6 . . U := { U j } is a finite open covering of M = N . Since we assume that ξ αj = z αj , α =1 , ..., n in subsection 6.2, the 1-cocycle ( { Λ ′ uj } , { θ ujk } ) ∈ C ( U , ∧ Θ M ) ⊕ C ( U , Θ M ) in (6 . . . .
17) represents the infinitesimal Poisson deformation (Λ ′ u , θ u ) = ϕ ( ∂∂t u ) ∈ H ( M , Θ • M )where ϕ is the Poisson Kodaira-Spencer map of the Poisson analytic family ( M , Λ M , B, ω ) (see HEOREM OF EXISTENCE AND COMPLETENESS FOR HOLOMORPHIC POISSON STRUCTURES 27
Proposition 3.1.7 and Definition 3.1.13). Since the coefficients Γ jkv ··· v l of the homogeneous polyno-mial Γ jk | v ( s ) = P v + ··· v l = v Γ jkv ··· v l s v · · · s v l l are holomorphic vector fields on U j ∩ U k , { Γ jk | v ( s ) } = P v + ··· v l = v { Γ jkv ··· v l } s v · · · s v l l is a homogenous polynomial of degree v whose coefficients are { Γ jkv ··· v l } ∈ C ( U , Θ M ). Since the coefficients λ jv ··· v l of the homogenous polynomial λ j | v ( s ) = P v + ··· + v l = v λ jv ··· v l s v · · · s v l l are holomorphic bivector fields on U j , { λ j | v ( s ) } = P v + ··· + v l = v { λ jv ··· v l } s v · · · s v l l is a homogenouspolynomial of degree v whose coefficients are { λ jv ··· v l } ∈ C ( U , ∧ Θ M ). Similarly { g j | v ( s ) } = P v + ··· + v l = v { g jv ··· v l } s v · · · s v l l is a homogenous polynomial of degree v whose coefficients are { g jv ··· v l } ∈ C ( U , Θ M ). We claim that Lemma 6.3.21.
The following equation holds ( { λ j | v ( s ) } , { Γ jk | v ( s ) } ) = m X u =1 h u | v ( s )( { Λ ′ uj } , { θ ujk } ) − δ HP ( { g j | v ( s ) } )(6.3.22) where δ HP ( { g j | v ( s ) } ) := ( − δ ( { g j | v } ) = { g j | v ( s ) − g k | v ( s ) } , { [ P nr,s =1 Λ r,sM j ( z j ) ∂∂z rj ∧ ∂∂z sj , g j | v ( s )] } ) .Here δ is the ˘Cech map.Proof. First, we have { Γ jk | v ( s ) } = P mu =1 h u | v ( s ) { θ rjk } + δ { g j | v ( s ) } (see [Kod05] p.291).It remains to show that { λ j | v ( s ) } = P mu =1 h u | v ( s ) { Λ ′ uj } − { [ P r,s Λ r,sM j ( z j ) ∂∂z rj ∧ ∂∂z sj , g j | v ( s )] } .Indeed, m X u =1 h u | v ( s )Λ ′ uj − n X r,s,β =1 [Λ r,sM j ( z j ) ∂∂z rj ∧ ∂∂z sj , g βj | v ( z j , s ) ∂∂z βj ]= m X u =1 h u | v ( s )Λ ′ uj − n X r,s,β =1 Λ r,sM j ∂g βj | v ∂z rj ∂∂z βj ∧ ∂∂z sj + n X r,s,β =1 g βj | v ∂ Λ r,sM j ∂z βj ∂∂z rj ∧ ∂∂z sj − n X r,s,β =1 Λ r,sM j ∂g βj | v ∂z sj ∂∂z rj ∧ ∂∂z βj = m X u =1 h u | v ( s )Λ ′ uj − n X r,s,β =1 Λ β,sM j ∂g rj | v ∂z βj ∂∂z rj ∧ ∂∂z sj + n X r,s,β =1 g βj | v ∂ Λ r,sM j ∂z βj ∂∂z rj ∧ ∂∂z sj − n X r,s,β =1 Λ r,βM j ∂g sj | v ∂z βj ∂∂z rj ∧ ∂∂z sj = λ j | v ( s )by (6 . . . .
17) and (6 . . (cid:3) Thus in order to construct h v ( s ) = h v − ( s ) + h v ( s ), g vj ( z j , s ) = g v − j ( z j , s ) + g j | v ( z j , s ) so that(6 . . v and (6 . . v hold, it suffices to obtain solutions h u | v , u = 1 , ..., m, { g j | v ( s ) } of the equations(6.3.22).If solutions h u | v ( s ) , u = 1 , ..., m, { g j | v ( s ) } exist, from (6.3.22), we have(6.3.23) [ P nr,s =1 Λ r,sM j ( z j ) ∂∂z rj ∧ ∂∂z sj , λ j | v ( s )] = 0 λ k | v ( s ) − λ j | v ( s ) + [ P nr,s =1 Λ r,sM j ( z j ) ∂∂z rj ∧ ∂∂z sj , Γ jk | v ( s )] = 0Γ jk | v ( s ) − Γ ik | v ( s ) + Γ ij | v ( s ) = 0Conversely, Lemma 6.3.24. If ( { λ j | v ( s ) } , { Γ jk | v ( s ) } ) satisfies (6 . . , then (6.3.25) ( { λ j | v ( s ) } , { Γ jk | v ( s ) } ) = m X u =1 h u | v ( s )( { Λ ′ uj } , { θ ujk } ) − δ HP ( { g j | v ( s ) } ) has solutions h u | v ( s ) , u = 1 , ..., m, { g j | v ( s ) } when the Poisson Kodaira-Spencer map ϕ : T ( B ) → H ( M , Θ • M ) is surjective.Proof. Let h u | v ( s ) = P v + ··· v l = v h uv ··· v l s v · · · s v l l . Then by considering the coefficients of s v · · · s v l l ,(6 . .
25) can be written as( { λ jv ··· v l } , { Γ jkv ,...,v l } ) = m X u =1 h uv ··· v l ( { Λ ′ uj } , { θ ujk } ) − δ HP ( { g jv ··· v l } ) . Thus it suffices to prove that any 1-cocycle ( { λ j } , { Γ jk } ) ∈ C ( U , ∧ Θ ) ⊕ C ( U , Θ ) such that[Λ , λ j ] = 0 , λ k − λ j + [Λ , Γ jk ] = 0 , Γ jk − Γ ik + Γ ij = 0 can be written in the form( { λ j } , { Γ jk } ) = m X u =1 h u ( { Λ ′ uj } , { θ ujk } ) − δ HP ( { g j } ) , for some h u ∈ C , { g j } ∈ C ( U , Θ )Let ( η, γ ) ∈ H ( M , Θ • M ) be the cohomology class of ( { λ j } , { Γ jk } ). Since ϕ : T ( B ) → H ( M , Θ • M )is surjective, ( η, γ ) is written in the form of a linear combination of the (Λ ′ u , θ u )(= the cohomologyclass of ( { Λ ′ uj } , { θ ujk } ), u = 1 , ..., m in (6 . . , (6 . . η, γ ) = m X u =1 h u (Λ ′ u , θ u ) , h u ∈ C So P mu =1 h u ( { Λ ′ uj } , { θ ujk } ) is cohomologous to ( { λ j } , { Γ jk } ). Therefore there exists { g j } ∈ C ( U , Θ )such that δ HP ( { g j } ) = P mu =1 h u ( { Λ ′ uj } , { θ ujk } ) − ( { λ j } , { Γ jk } ) . (cid:3) Next we will prove that
Lemma 6.3.26. ( { λ j | v ( s ) } , { Γ jk | v ( s ) } ) satisfies (6 . . .Proof. First, we have Γ jk | v ( s ) − Γ ik | v ( s ) + Γ ij | v ( s ) = 0 (see [Kod05] p.292). Second, we showthat [Λ , λ j | v ( s )] = 0. We note that for Π ∈ Γ( U j , ∧ Θ M ), Π = 0 if and only if Π( z aj , z bj , z cj ) :=Π( dz aj ∧ dz bj ∧ dz cj ) for any a, b, c . Then from [Λ N j , Λ N j ] = 0, [Λ M j , Λ M j ] = 0, and Lemma 4.3.5, [Λ , λ j | v ]( z aj , z bj , z cj )= Λ ( λ j | v ( z aj , z bj ) , z cj ) − Λ ( λ j | v ( z aj , z cj ) , z bj ) + Λ ( λ j | v ( z bj , z cj ) , z aj )+ λ j | v (Λ ( z aj , z bj ) , z cj ) − λ j | v (Λ ( z aj , z cj ) , z bj ) + λ j | v (Λ ( z bj , z cj ) , z aj ) ≡ v Λ N j ( λ j | v ( z aj , z bj ) , g cv − j ) − Λ N j ( λ j | v ( z aj , z cj ) , g bv − j ) + Λ N j ( λ j | v ( z bj , z cj ) , g cv − j )+ λ j | v (Λ M j ( z aj , z bj ) , z cj ) − λ j | v (Λ M j ( z aj , z cj ) , z bj ) + λ j | v (Λ M j ( z bj , z cj ) , z aj ) ≡ v − N j (Λ a,bM j ( g v − j , h v − ) , g cv − j ) + Λ N j (Λ N j ( g av − j , g bv − j ) , g cv − j )+ 2Λ N j (Λ a,cM j ( g v − j , h v − ) , g bv − j ) − Λ N j (Λ N j ( g av − j , g cv − j ) , g bv − j ) − N j (Λ b,cM j ( g v − j , h v − ) , g av − j ) + Λ N j (Λ N j ( g bv − j , g cv − j ) , g av − j )+ 2 λ j | v (Λ a,bM j ( z j , s ) , z cj ) − λ j | v (Λ a,cM j ( z j , s ) , z bj ) + 2 λ j | v (Λ b,cM j ( z j , s ) , z aj ) ≡ v − N j (Λ a,bM j ( g v − j , h v − ) , g cv − j ) + 2Λ N j (Λ a,cM j ( g v − j , h v − ) , g bv − j ) − N j (Λ b,cM j ( g v − j , h v − ) , g av − j )+ 4 n X r =1 − Λ r,cM j ( g v − j , h v − ) + n X α,β =1 Λ α,βN j ( z j , s ) ∂g rv − j ∂z αj ∂g cv − j ∂z βj ∂ Λ a,bM j ∂z rj ( g v − j , h v − ) − n X r =1 − Λ r,bM j ( g v − j , h v − ) + n X α,β =1 Λ α,βN j ( z j , s ) ∂g rv − j ∂z αj ∂g bv − j ∂z βj ∂ Λ a,cM j ∂z rj ( g v − j , h v − )+ 4 n X r =1 − Λ r,aM j ( g v − j , h v − ) + n X α,β =1 Λ α,βN j ( z j , s ) ∂g rv − j ∂z αj ∂g av − j ∂z βj ∂ Λ b,cM j ∂z rj ( g v − j , h v − ) ≡ v − N j (Λ a,bM j ( g v − j , h v − ) , g cv − j ) + 2Λ N j (Λ a,cM j ( g v − j , h v − ) , g bv − j ) − N j (Λ b,cM j ( g v − j , h v − ) , g av − j ) − n X r =1 Λ r,cM j ( g v − j , h v − ) ∂ Λ a,bM j ∂z rj ( g v − j , h v − ) − Λ r,bM j ( g v − j , h v − ) ∂ Λ a,cM j ∂z rj ( g v − j , h v − ) + Λ r,aM j ( g v − j , h v − ) ∂ Λ b,cM j ∂z rj ( g v − j , h v − ) ! + 4 n X α,β =1 Λ α,βN j ( z j , s ) ∂ Λ a,bM j ( g v − j , h v − ) ∂z αj ∂g cv − j ∂z βj − n X α,β =1 Λ α,βN j ( z j , s ) ∂ Λ a,cM j ( g v − j , h v − ) ∂z αj ∂g bv − j ∂z βj + 4 n X α,β =1 Λ α,βN j ( z j , s ) ∂ Λ b,cM j ( g v − j , h v − ) ∂z αj ∂g av − j ∂z βj = 0 HEOREM OF EXISTENCE AND COMPLETENESS FOR HOLOMORPHIC POISSON STRUCTURES 29
Next we will show that λ k | v ( s ) − λ j | v ( s ) + [ n X r,s =1 Λ r,sM j ( z j ) ∂∂z rj ∧ ∂∂z sj , Γ jk | v ( s )] = 0 . (6.3.27)First we compute the third term of (6.3.27)[ n X r,s =1 Λ r,sM j ( z j ) ∂∂z rj ∧ ∂∂z sj , Γ jk | v ( s )] = n X r,s,β =1 [Λ r,sM j ( z j ) ∂∂z rj ∧ ∂∂z sj , Γ βjk | v ∂∂z βj ](6.3.28) = n X r,s,β =1 Λ r,sM j ∂ Γ βjk ∂z rj ∂∂z βj ∧ ∂∂z sj − Γ βjk | v ∂ Λ r,sM j ∂z βj ∂∂z rj ∧ ∂∂z sj + Λ r,sM j ∂ Γ βjk | v ∂z sj ∂∂z rj ∧ ∂∂z βj = n X r,s,β =1 Λ β,sM j ∂ Γ rjk ∂z βj ∂∂z rj ∧ ∂∂z sj − Γ βjk | v ∂ Λ r,sM j ∂z βj ∂∂z rj ∧ ∂∂z sj + Λ r,βM j ∂ Γ sjk | v ∂z βj ∂∂z rj ∧ ∂∂z sj ! We consider the first term λ k | v ( s ) of (6 . . . .
14) and (6 . . λ k | v ( s ) ≡ v n X p,q =1 − Λ p,qM k ( g v − k ( z k , s ) , h v − ( s )) + n X a,b =1 Λ a,bN k ( z k , s ) ∂g pkv − ∂z ak ∂g qkv − ∂z bk ∂∂z pk ∧ ∂∂z qk (6.3.29) ≡ v n X r,s =1 n X p,q =1 − Λ p,qM k ( g v − k ( z k , s ) , h v − ( s )) + n X a,b =1 Λ a,bN k ( z k , s ) ∂g pkv − ∂z ak ∂g qkv − ∂z bk ∂b rjk ∂z pk ∂b sjk ∂z qk ∂∂z rj ∧ ∂∂z sj We consider the second term − λ j | v ( s ) of (6.3.27). We note that since z j = b jk ( z k ) = f jk ( z k , . . λ j | v ( z j , s ) ≡ v λ j | v ( f jk ( z k , s ) , s ). Then from (6 . .
14) and by induction hypoth-esis (6 . . − λ j | v ( z j , s ) ≡ v n X r,s =1 Λ r,sM j ( g v − j ( z j , s ) , h v − ( s )) − n X α,β =1 Λ α,βN j ( z j , s ) ∂g rj v − ∂z αj ∂g sj v − ∂z βj ∂∂z rj ∧ ∂∂z sj (6.3.30) ≡ v n X r,s =1 Λ r,sM j ( g v − j ( f jk ( z k , s ) , s ) , h v − ( s )) − n X α,β =1 Λ α,βN j ( f jk ( z k , s ) , s ) ∂g rj v − ∂z αj ( f jk ( z k , s ) , s ) ∂g sj v − ∂z βj ( f jk ( z k , s ) , s ) ∂∂z rj ∧ ∂∂z sj We consider the first term of (6 . . . .
7) and (6 . . r,sM j ( g v − j ( f jk ( z k , s ) , s ) , h v − ( s )) ≡ v Λ r,sM j ( g jk ( g v − k ( z k , s ) , h v − ( s )) + Γ jk | v ( z j , s ) , h v − ( s ))(6.3.31) ≡ v Λ r,sM j ( g jk ( g v − k ( z k , s ) , h v − ( s )) , h v − ( s )) + n X β =1 ∂ Λ r,sM j ∂z βj Γ βjk | v ( z j , s )= n X p,q =1 Λ p,qM k ( g v − k ( z k , s ) , h v − ( s )) ∂g rjk ∂ξ pk ( g v − k ( z k , s ) , h v − ( s )) ∂g sjk ∂ξ qk ( g v − k ( z k , s ) , h v − ( s )) + n X β =1 ∂ Λ r,sM j ∂z βj Γ βjk | v ( z j , s ) On the other hand, we consider the second term of (6 . . . . . .
7) and(6 . . n X α,β =1 Λ α,βN j ( f jk ( z k , s ) , s ) ∂g rj v − ∂z αj ( f jk ( z k , s ) , s ) ∂g sj v − ∂z βj ( f jk ( z k , s ) , s )(6.3.32)= n X α,β,a,b =1 Λ a,bN k ( z k , s ) ∂f αjk ( z k , s ) ∂z ak ∂f βjk ( z k , s ) ∂z bk ∂g rj v − ∂z αj ( f jk ( z k , s ) , s ) ∂g sj v − ∂z βj ( f jk ( z k , s ) , s )= n X a,b =1 Λ a,bN k ( z k , s ) ∂ ( g rv − j ( f jk ( z k , s ) , s )) ∂z ak ∂ ( g sv − j ( f jk ( z k , s ) , s )) ∂z bk ≡ v n X a,b =1 Λ a,bN k ( z k , s ) ∂ ( g rjk ( g v − k ( z k , s ) , h v − ( s )) + Γ rjk | v ( z j , s )) ∂z ak ∂ ( g sjk ( g v − k ( z k , s ) , h v − ( s )) + Γ sjk | v ( z j , s )) ∂z bk ≡ v n X a,b,p,q =1 Λ a,bN k ( z k , s ) ∂g rjk ∂ξ pk ∂g pv − k ∂z ak ∂g sjk ∂ξ qk ∂g qv − k ∂z bk + n X a,b =1 Λ a,bM k ( z k ) ∂ Γ rjk | v ∂z ak ∂z sj ∂z bk + n X a,b =1 Λ a,bM k ( z k ) ∂z rj ∂z ak ∂ Γ sjk | v ∂z bk ≡ v n X a,b,p,q =1 Λ a,bN k ( z k , s ) ∂g rjk ∂ξ pk ∂g pv − k ∂z ak ∂g sjk ∂ξ qk ∂g qv − k ∂z bk + n X a,b,β =1 Λ a,bM k ( z k ) ∂ Γ rjk | v ∂z βj ∂z βj ∂z ak ∂z sj ∂z bk + n X a,b,β =1 Λ a,bM k ( z k ) ∂z rj ∂z ak ∂ Γ sjk | v ∂z βj ∂z βj ∂z bk ≡ v n X a,b,p,q =1 Λ a,bN k ( z k , s ) ∂g rjk ∂ξ pk ∂g pv − k ∂z ak ∂g sjk ∂ξ qk ∂g qv − k ∂z bk + n X β =1 Λ β,sM j ( z j ) ∂ Γ rjk | v ∂z βj + n X β =1 Λ r,βM j ( z j ) ∂ Γ sjk | v ∂z βj where we mean ∂g rjk ∂ξ pk and ∂g sjk ∂ξ qk by ∂g rjk ∂ξ pk := ∂g rjk ∂ξ pk ( g v − k ( z k , s ) , h v − ( s )) , ∂g sjk ∂ξ qk := ∂g sjk ∂ξ qk ( g v − k ( z k , s ) , h v − ( s ))(6.3.33)Hence from (6.3.30),(6.3.31), and (6.3.32), we have − λ j | v ( s ) ≡ v n X r,s =1 n X p,q =1 (Λ p,qM k ( g v − k ( z k , s ) , h v − ( s )) ∂g rjk ∂ξ pk ∂g sjk ∂ξ qk + n X β =1 ∂ Λ r,sM j ∂z βj Γ βjk | v ( z j , s ) ∂∂z rj ∧ ∂∂z sj (6.3.34)+ n X r,s =1 − n X a,b,p,q =1 Λ a,bN k ( z k , s ) ∂g rjk ∂ξ pk ∂g pv − k ∂z ak ∂g sjk ∂ξ qk ∂g qv − k ∂z bk − n X β =1 Λ β,sM j ( z j ) ∂ Γ rjk | v ∂z βj − n X β =1 Λ r,βM j ( z j ) ∂ Γ sjk | v ∂z βj ∂∂z rj ∧ ∂∂z sj From (6.3.28),(6.3.29) and (6 . . r, s , n X p,q =1 − Λ p,qM k ( g v − k ( z k , s ) , h v − ( s )) + n X a,b =1 Λ a,bN k ( z k , s ) ∂g pkv − ∂z ak ∂g qkv − ∂z bk ∂b rjk ∂z pk ∂b sjk ∂z qk (6.3.35)+ n X p,q =1 Λ p,qM k ( g v − k ( z k , s ) , h v − ( s )) ∂g rjk ∂ξ pk ∂g sjk ∂ξ qk − n X a,b,p,q =1 Λ a,bN k ( z k , s ) ∂g rjk ∂ξ pk ∂g pv − k ∂z ak ∂g sjk ∂ξ qk ∂g qv − k ∂z bk ≡ v . .
35) is equivalent to n X p,q =1 − Λ p,qM k ( g v − k ( z k , s ) , h v − ( s )) + n X a,b =1 Λ a,bN k ( z k , s ) ∂g pkv − ∂z ak ∂g qkv − ∂z bk (cid:18) ∂b rjk ∂z pk ∂b sjk ∂z qk − ∂g rjk ∂ξ pk ∂g sjk ∂ξ qk (cid:19) ≡ v HEOREM OF EXISTENCE AND COMPLETENESS FOR HOLOMORPHIC POISSON STRUCTURES 31
By induction hypothesis (6 . . p,qM k ( g v − k ( z k , s ) , h v ( s )) ≡ v − P na,b =1 Λ a,bN k ( z k , s ) ∂g pkv − ∂z ak ∂g qkv − ∂z bk ,and we have (cid:16) ∂b rjk ∂z pk ∂b sjk ∂z qk − ∂g rjk ∂ξ pk ∂g sjk ∂ξ qk (cid:17) ≡ . . ∂g rjk ∂ξ pk ( g v − k ( z k , , h v − (0)) = ∂g rjk ∂ξ pk ( z k ,
0) = ∂b rjk ∂z pk and similarly for ∂g sjk ∂ξ qk . Hence we have (6.3.36). This completes Lemma 6.3.26. (cid:3) Proof of Convergence.
By Lemma 6.3.24 and Lemma 6.3.26, we can find h u | v , u = 1 , ..., m , and { g j | v ( z j , s ) } inductivelyon v such that h v ( s ) = h v − ( s ) + h u | v ( s ) and g vj ( z j , s ) = g v − j ( z j , s ) + g j | v ( z j , s ) satisfy (6 . . v and(6 . . v so that we have formal power series h ( s ) and g j ( z j , s ) satisfying (6.3.1) and (6.3.2). In thissubsection, we will prove that we can choose appropriate solutions h u | v ( s ) and { g j | v ( z j , s ) } in eachinductive step so that h ( s ) and g j ( z j , s ) converge absolutely in | s | < ǫ if ǫ > jk | v ( z j , s ), λ j | v ( s ) and use Lemma 6.4.10below concerning the “magnitude” of the solutions h u | v ( s ) , u = 1 ..., m, { g j | v ( z j , s ) } of the equation(6 . . Definition 6.4.1.
Let U := { U j } be a finite open covering of M in (6 . . . We may assume that U j = { z j ∈ C n | z j | < } and M = S j U δj , where U δj = { z j ∈ U j || z j | < − δ } for a sufficiently smallnumber δ > . We denote a -cocycle ( { λ j } , { Γ jk } ) ∈ C ( U , ∧ Θ M ) ⊕ C ( U , Θ M ) by ( λ, Γ) in the˘Cech resolution of the complex of sheaves (3 . . , and define its norm by | ( λ, Γ) | := max j sup z j ∈ U δj | λ j ( z j ) | + max j,k sup z j ∈ U j ∩ U k | Γ jk ( z j ) | (6.4.2) Remark 6.4.3.
We explain the meaning of | λ j ( z j ) | and | Γ jk ( z j ) | in (6 . . . We regard holomorphicvector field Γ jk ( z j ) = P nα =1 Γ αjk ( z j ) ∂∂z αj as a vector-valued holomorphic function (Γ jk ( z j ) , ..., Γ njk ( z j )) and regard holomorphic bivector field λ j ( z j ) = P nr,s =1 λ r,sj ( z j ) ∂∂z rj ∧ ∂∂z sj as a holomorphic vectorvalued function ( λ , j ( z j ) , · · · , λ r,sj ( z j ) , · · · , λ n,nj ( z j )) . For z j ∈ U j ∩ U k , we define | Γ jk ( z j ) | :=max α | Γ αjk ( z j ) | . On the other hand, for z j ∈ U δj we define | λ j ( z j ) | := max r,s | λ r,sj ( z j ) | . Remark 6.4.4.
Since each U j = { z j ∈ C n || z j | < } in (6 . . is a coordinate polydisk, we mayassume that the coordinate function z j is defined on a domain of M containing U j ( the closure of U j ) . Hence there exists a constant L > such that for all α, β = 1 , ..., n , and for all U k ∩ U j = ∅ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂z αj ∂z βk ( z k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂b αjk ∂z βk ( z k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < L , z k ∈ U k ∩ U j , (6.4.5) and there exist constants C, C ′ > such that for all r, s, β = 1 , ..., n and for all U j , (cid:12)(cid:12)(cid:12) Λ r,sM j ( z j ) (cid:12)(cid:12)(cid:12) < C, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ Λ r,sM j ∂z βj ( z j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < C ′ , z j ∈ U j . (6.4.6) We define the norm of the matrix B jk ( z j ) := (cid:18) ∂z αj ∂z βk ( z k ) (cid:19) α,β =1 ,...,n by | B jk ( z k ) | = max α P β (cid:12)(cid:12)(cid:12)(cid:12) ∂z αj ∂z βk ( z k ) (cid:12)(cid:12)(cid:12)(cid:12) .Then there exists a constant K > such that for all U j ∩ U k = ∅ , | B jk ( z k ) | < K z k ∈ U k ∩ U j (6.4.7) Since θ αujk ( z j ) = (cid:16) ∂g αjk ( z k ,t ) ∂t u (cid:17) t =0 in (6 . . are bounded on U j ∩ U k = ∅ , there exists a constant K such that | θ ujk ( z j ) | = | n X α =1 θ αujk ( z j ) ∂∂z αj | := max α | θ αujk ( z j ) | < K (6.4.8) Since Λ ′ r,suj ( z j ) = (cid:18) ∂ Λ r,sMj ( z j ,t ) ∂t u (cid:19) t =0 in (6 . . are bounded on U j , there exists a constant L suchthat | Λ ′ uj ( z j ) | = | n X r,s =1 Λ ′ r,suj ( z j ) ∂∂z rj ∧ ∂∂z sj | := max r,s | Λ ′ r,suj ( z j ) | < L (6.4.9) Lemma 6.4.10 (compare [Kod05] Lemma 6.2 p.295) . There exist solutions h u , u = 1 , ..., m , and { g j ( z j ) = P nα =1 g αj ( z j ) ∂∂z αj } of the equation ( { λ j } , { Γ jk } ) = m X u =1 h u ( { Λ ′ uj } , { θ ujk } ) − δ HP { g j ( z j ) } (6.4.11) which satisfy | h u | ≤ M | ( λ, Γ) | , | g j ( z j ) | := max α | g αj ( z j ) | ≤ M | ( λ, Γ) | for z j ∈ U j where M is a constant independent of a -cocycle ( λ, Γ) = ( { λ j } , { Γ jk } ) .Proof. The proof is similar to [Kod05] Lemma 6.2 p.295 to which we refer for the detail. For a1-cocycle ( λ, Γ) = ( { λ j } , { Γ jk } ) with | ( λ, Γ) | < ∞ , we define ι ( λ, Γ) by ι ( λ, Γ) = inf max u,j {| h u | , sup z j ∈ U j | g j ( z j ) |} , where inf is taken with respect to all the solutions h u , u = 1 , ..., m , and g j ( z j ) of (6.4.11). We willshow that there exists a constant M such that for all 1-cocycles ( λ, Γ) ∈ C ( U , ∧ Θ M ) ⊕C ( U , Θ M ),we have ι ( λ, Γ) ≤ M | ( λ, Γ) | Suppose there is no such constant M . Then we can find a sequence of 1-cocycles ( λ ( v ) , Γ ( v ) ) =( { λ ( v ) j } , { Γ ( v ) jk } ) ∈ C ( U , ∧ Θ M ) ⊕ C ( U , Θ M ) , v = 1 , , , · · · , and their solutions g ( v ) j ( z j ) , h ( v ) u , u =1 , ..., m such that ι ( λ ( v ) , Γ ( v ) ) = 1 , | ( λ ( v ) , Γ ( v ) ) | < v , (6.4.12)and the sequence { h ( v ) u } , u = 1 , ..., m converge and { g ( v ) j ( z j ) } converges uniformly on U j .Put h u = lim v →∞ h ( v ) u , and g j ( z j ) = lim v →∞ g ( v ) j ( z j ) and note thatΓ ( v ) jk ( z j ) = m X u =1 h ( v ) u θ ujk ( z j ) + B jk ( z k ) g ( v ) k ( z k ) − g ( v ) j ( z j ) ,λ ( v ) j ( z j ) = m X u =1 h ( v ) u Λ ′ uj ( z j ) − [ n X r,s =1 Λ r,sM j ( z j ) ∂∂z rj ∧ ∂∂z sj , n X α =1 g α ( v ) j ( z j ) ∂∂z αj ]Since | Γ ( v ) jk ( z j ) | ≤ | ( λ ( v ) , Γ ( v ) ) | → z j ∈ U j ∩ U k , and | λ ( v ) j ( z j ) | ≤ | ( λ ( v ) , Γ ( v ) ) | → z j ∈ U δj as v → ∞ , we have0 = m X u =1 h u θ ujk ( z j ) + B jk ( z k ) g k ( z k ) − g j ( z j ) , z j ∈ U j ∩ U k m X u =1 h u Λ ′ uj ( z j ) − [ n X r,s =1 Λ r,sM j ( z j ) ∂∂z rj ∧ ∂∂z sj , n X α =1 g αj ( z j ) ∂∂z αj ] , z j ∈ U δj By identity theorem, we have0 = m X u =1 h u Λ ′ uj ( z j ) − [ n X r,s =1 Λ r,sM j ( z j ) ∂∂z rj ∧ ∂∂z sj , n X α =1 g αj ( z j ) ∂∂z αj ] , z j ∈ U j HEOREM OF EXISTENCE AND COMPLETENESS FOR HOLOMORPHIC POISSON STRUCTURES 33
By putting ˜ h ( v ) u = h ( v ) u − h u , and ˜ g ( v ) j ( z j ) = g ( v ) j ( z j ) − g j ( z j ), we obtainΓ ( v ) jk ( z j ) = m X u =1 ˜ h ( v ) u θ ujk ( z j ) + B jk ( z k )˜ g ( v ) k ( z k ) − ˜ g ( v ) j ( z j ) ,λ ( v ) j ( z j ) = m X u =1 ˜ h ( v ) u Λ ′ uj ( z j ) − [ n X r,s =1 Λ r,sM j ( z j ) ∂∂z rj ∧ ∂∂z sj , n X α =1 ˜ g α ( v ) j ( z j ) ∂∂z αj ]Hence ˜ h ( v ) u , u = 1 , ..., m , and { ˜ g ( v ) j ( z j ) } satisfy the equation (6.4.11) for ( λ, Γ) = ( { λ ( v ) } , { Γ ( v ) } ).This is a contradiction to ι ( λ, Γ) = 1 ((6.4.12)) since we have ˜ h ( v ) u → z j ∈ U j | ˜ g ( v ) j ( z j ) | → (cid:3) Next we will prove that we can choose appropriate solutions h u | v ( s ) , u = 1 , ..., m and { g j | v ( z j , s ) } in each inductive step by estimating Γ jk | v ( z j , s ) , λ j | k ( z j , s ) and using Lemma 6.4.10 so that theformal power series h ( s ) and g j ( z j , s ) converge absolutely in | s | < ǫ if ǫ > Remark 6.4.13. (1)
For two power series of s , ..., s l , P ( s ) = ∞ X v ,...,v l =0 P v ,...,v l s v · · · s v l l , P v ,...,v l ∈ C n ,a ( s ) = ∞ X v ,...,v l =0 a v ,...,v l s v · · · s v l l , , a v ,...,v l ≥ , we write P ( s ) ≪ a ( s ) if | P v ,...,v l | ≤ a v ,...,v l , v , ..., v l = 0 , , , ... . (2) For a power series P ( s ) , we denote by [ P ( s )] v the term of homogeneous part of degree v with respect to s . (3) For A ( s ) = b c P ∞ v =1 c v ( s + ··· s l ) v v , b > , c > , we have A ( s ) v ≪ (cid:0) bc (cid:1) v − A ( s ) , v = 2 , , ... (4) Recall that for each U j = { z j ∈ C n || z j | < } , we set U δj = { z j ∈ U j || z j | < − δ } for a given δ . Then M = S j U δj for a sufficiently small δ . To prove the convergence of h ( s ) and g j ( z j , s ), we will show the estimates h ( s ) ≪ A ( s ) , g j ( z j , s ) − z j ≪ A ( s ) for suitable constants b and c in Remark 6.4.13 (3), equivalently h v ( s ) ≪ A ( s ) , g vj ( z j , s ) − z j ≪ A ( s )(6.4.14)for v = 1 , , , ... . We will prove this by induction on v = 1 , , , ... . For v = 1, since the linear termof A ( s ) is b ( s + · · · + s l ), the estimate holds if b is sufficiently large. Let v ≥ v −
1. In other words, h v − ( s ) ≪ A ( s ) , g v − j ( z j , s ) − z j ≪ A ( s )(6.4.15)We will prove that (6.4.14) holds. For this, we estimate Γ jk | v ( z j , s ) and λ j | v ( z j , s ). For the esti-mation of Γ jk | v ( z j , s ), we briefly summarize Kodaira’s estimation presented in [Kod05] p.298-302in the following: since f jk ( z k , s ) = b jk ( z k ) + P ∞ v =1 f jk | v ( z j , s ) are given vector-valued holomorphicfunctions, we may assume that f jk ( z k , s ) − b jk ( z k ) ≪ A ( s ) , A ( s ) = b c ∞ X v =1 c v ( s + · · · + s l ) v v holds for z k ∈ U k ∩ U j with b > c > b c δ < , where δ from Remark 6.4.13 (4).If we take b and c such that b > b , c > c , ba ( m + n ) c < , we can estimateΓ jk | v ( z j , s ) ≪ K K ∗ A ( s ) , z j ∈ U j ∩ U k . (6.4.16)where K ∗ = n +1 b cδ + b b + ba ( m + n ) c and K from (6.4.7) (for the detail, see page [Kod05] 298-302). Next we estimate λ j | v ( z j , s ) (see (6.3.20)). To estimate it, we estimate λ r,sj | v ( z j , s ) for each pair( r, s ) where r, s = 1 , ..., n . We note that from (6.3.14), we have λ r,sj | v ( z j , s ) = [ − Λ r,sM j ( g v − j ( z j , s ) , h v − ( s ))] v + [ n X p,q =1 Λ p,qN j ( z j , s ) ∂g rj v − ∂z pj ∂g sj v − ∂z qj ] v (6.4.17)First we estimate [Λ r,sM j ( g v − j ( z j , s ) , h v − ( s ))] v in (6 . . r,sM j ( z j + ξ, t ) into powerseries in ξ , ..., ξ n , t , ..., t m , and let L ( ξ, t ) be its linear term. Since Λ r,sM j ( z j ,
0) = Λ r,sM j ( z j ) from(6 . . r, s ),Λ r,sM j ( z j + ξ, t ) − Λ r,sM j ( z j ) − L ( ξ, t ) ≪ ∞ X µ =2 d µ ( ξ + · · · + ξ n + t + · · · + t m ) µ for some constant d > ξ = g v − j ( z j , s ) − z j , and t = h v − ( s ). Since ξ ≪ A ( s ) and t ≪ A ( s ) by induction hypothesis(6.4.15), we have from Remark 6.4.13 (3)Λ r,sM j ( g v − j ( z j , s ) , h v − ( s )) − Λ r,sM j ( z j ) − L ( g v − j ( z j , s ) − z j , h v − ( s )) ≪ ∞ X µ =2 d µ ( n + m ) µ A ( s ) µ ≪ ∞ X µ =2 d µ ( m + n ) µ (cid:18) bc (cid:19) µ − A ( s ) = bd ( m + n ) c ∞ X µ =2 (cid:18) bd ( m + n ) c (cid:19) µ − A ( s )Hence we have[Λ r,sM j ( g v − j ( z j , s ) , h v − ( s ))] v ≪ bd ( m + n ) c ∞ X µ =0 (cid:18) bd ( m + n ) c (cid:19) µ A ( s )Choose a constant c such that bd ( m + n ) c < . Then we have[Λ r,sM j ( g v − j ( z j , s ) , h v − ( s ))] v ≪ bd ( m + n ) c A ( s ) , z j ∈ U j (6.4.18)Next we estimate [ P np,q =1 Λ p,qN j ( z j , s ) ∂g rv − j ( z j ,s ) ∂z pj ∂g sv − j ( z j ,s ) ∂z qj ] v in (6 . . α rv − j ( z j , s ) := g rv − j ( z j , s ) − z rj ≪ A ( s ) for all r = 1 , ..., n (6.4.19)Since Λ p,qN j ( z j , s ) is holomorphic, and Λ p,qN j ( z j ,
0) = Λ p,qM j ( z j ) from (6 . . p, q ),Π p,qj ( z j , s ) := Λ p,qN j ( z j , s ) − Λ p,qM j ( z j ) ≪ A ( s ) = b c ∞ X v =1 c v ( s + · · · + s l ) v v (6.4.20)for some constants b , c >
0. If we choose b > b and c > c , then we haveΠ p,qj ( z j , s ) ≪ b b A ( s ) . (6.4.21)Now assume that z j = ( z j , ..., z nj ) ∈ U δj from Remark 6.4.13 (4). Then by Cauchy’s integral formula,and (6 . . p = 1 , ..., n , ∂α rv − j ( z j , s ) ∂z pj = 12 πi Z | ξ − z pj | = δ α rv − j ( z j , ..., p -th ξ , ..., z nj , s )( ξ − z pj ) dξ Hence we have, for p = 1 , ..., n, ∂α rv − j ( z j , s ) ∂z pj ≪ A ( s ) δ (6.4.22) HEOREM OF EXISTENCE AND COMPLETENESS FOR HOLOMORPHIC POISSON STRUCTURES 35
Then from (6.4.19),(6.4.20), (6.4.21), (6.4.22) and (6 . . n X p,q =1 Λ p,qN j ( z j , s ) ∂g rv − j ( z j , s ) ∂z pj ∂g sv − j ( z j , s ) ∂z qj ] v (6.4.23)= [ n X p,q =1 (Π p,qj ( z j , s ) + Λ p,qM j ( z j )) ∂ ( α rv − j ( z j , s ) + z rj ) ∂z pj ∂ ( α sv − j ( z j , s ) + z sj ) ∂z qj ] v = [ n X p,q =1 Π p,qj ( z j , s ) ∂α rv − j ( z j , s ) ∂z pj ∂α sv − j ( z j , s ) ∂z qj ] v + [ n X p,q =1 Π p,qj ( z, s ) ∂α rv − j ( z, s ) ∂z pj ∂z sj ∂z qj ] v + [ n X p,q =1 Π p,qj ( z j , s ) ∂z rj ∂z pj ∂α sv − j ( z j , s ) ∂z qj ] v + [ n X p,q =1 Π p,qj ( z j , s ) ∂z rj ∂z pj ∂z sj ∂z qj ] v + [ n X p,q =1 Λ p,qM j ( z j ) ∂α rv − j ( z, s ) ∂z pj ∂α sv − j ( z j , s ) ∂z qj ] v ≪ n b bδ A ( s ) + 2 nb bδ A ( s ) + b b A ( s ) + Cn δ A ( s ) for C > b, c. ≪ (cid:18) n b b bc δ + 2 nb bbcδ + b b + Cn bcδ (cid:19) A ( s ) = (cid:18) n b bc δ + 2 nb cδ + b b + Cn bcδ (cid:19) A ( s ) from Remark 6.4.13 (3)Hence from (6.4.17),(6.4.18), (6.4.23), we have λ r,sj | v ( z j , s ) ≪ LA ( s ) , z j ∈ U δj (6.4.24)where L = bd ( m + n ) c + n b bc δ + nb cδ + b b + Cn bcδ .Then by Lemma 6.4.10, (6 . .
16) and (6 . . h u | v ( s ), u = 1 , ..., m , { g j | v ( s ) } such that h u | v ( s ) ≪ N A ( s ) , g j | v ( s ) ≪ N A ( s ) , where N = M (2 K K ∗ + L )Note that N is independent of v and K ∗ = n +1 b cδ + b b + ba ( m + n ) c , L = bd ( m + n ) c + n b bc δ + nb cδ + b b + Cn bcδ . If we first choose a sufficiently large b , and then choose c so that cb be sufficiently large(so that bc is sufficiently small and bc is sufficiently small), then we obtain N ≤
1. Note that b and c satisfy b > max { b , b } , c > max { c , c } , ba ( m + n ) c < and bd ( m + n ) c < .Hence the above solution h u | v ( s ) , u = 1 , ..., m, { g j | v ( s ) } satisfy the inequalities h v ( s ) ≪ A ( s ) , g j | v ( s ) ≪ A ( s )Since h v ( s ) = h v − ( s ) + h v ( s ) , g vj ( z j , s ) = g v − j ( z j , s ) + g j | v ( z j , s ), we have h v ( s ) ≪ A ( s ), and g vj ( z j , s ) ≪ A ( s ). This completes the induction, and so we have h ( s ) ≪ A ( s ) and g j ( z j , s ) − z j ≪ A ( s ). These inequalities imply that, if | s | < lc , h ( s ) converges absolutely, and g j ( z j , s ) convergesabsolutely and uniformly for z j ∈ U j .6.5. Proof of Theorem 6.1.4.
By the same argument presented in page 303-304, we can glue together each g j on U δj × ∆ ǫ to construct a Poisson holomorphic map g : π − (∆ ǫ ) = ( S j U δj × ∆ ǫ , Λ N | ∆ ǫ ) → ( M , Λ M ) whichextends the identity map g : ( N , Λ ) → ( M = N , Λ ) (see [Kod05] page 303-304 for the detailand notations). This completes the proof of Theorem 6 . . Example 6.5.1.
Let U i = { [ z , z , z ] | z i = 0 } i = 0 , , be an open cover of complex projectiveplane P C . Let x = z z and w = z z be coordinates on U . Then the holomorphic Poisson structureson U are parametrized by t = ( t , ..., t ) ∈ C ( t + t x + t w + t x + t xw + t w + t x + t x w + t xw + t w ) ∂∂x ∧ ∂∂w This parametrizes the whole holomorphic Poisson structures on P C ( see [HX11] Proposition 2.2 ) .Let Λ = x ∂∂x ∧ ∂∂w be the holomorphic Poisson structure on P C . Then H ( P C , Θ • P C ) = 5 , H ( P C , Θ • P C ) =0( see [HX11] Example . . w ∂∂x ∧ ∂∂w , x ∂∂x ∧ ∂∂w , x w ∂∂x ∧ ∂∂w , xw ∂∂x ∧ ∂∂w and w ∂∂x ∧ ∂∂w are the representatives of the cohomology classes consisting of the basis of H ( P C , Θ P C ) . Let t = ( t , t , t , t , t ) ∈ C . Let Λ( t ) = ( t w + x + t x + t x w + t xw + t w ) ∂∂x ∧ ∂∂w be theholomorphic Poisson structure on P C × C . Then ( P C × C , Λ( t ) , C , ω ) , where ω is the naturalprojection, is a Poisson analytic family with ω − (0) = ( P C , Λ ) . Since the complex structure doesnot change in the family, the Poisson Kodaira-Spencer map is an isomorphism. Hence the Poissonanalytic family is complete at . References [ELW99] Sam Evens, Jiang-Hua Lu, and Alan Weinstein,
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