Deformations of compact holomorphic Poisson submanifolds
aa r X i v : . [ m a t h . AG ] A ug DEFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS
CHUNGHOON KIM
Abstract.
In this paper, we study deformations of compact holomorphic Poisson submanifolds which ex-tend Kodaira’s series of papers on semi-regularity (deformations of compact complex submanifolds of codi-mension 1) ([KS59]), deformations of compact complex submanifolds of arbitrary codimensions ([Kod62]),and stability of compact complex submanifolds ([Kod63]) in the context of holomorphic Poisson deforma-tions. We also study simultaneous deformations of holomorphic Poisson structures and holomorphic Poissonsubmanifolds on a fixed underlying compact complex manifold. In appendices, we present deformationsof Poisson closed subschemes in the language of functors of Artin rings which is the algebraic version ofdeformations of holomorphic Poisson submanifolds. We identify first-order deformations and obstructions.
Contents
1. Introduction 12. Deformations of compact holomorphic Poisson submanifolds of codimension 1 and Poissonsemi-regularity 73. Deformations of compact holomorphic Poisson submanifolds of arbitrary codimensions 174. Simultaneous deformations of holomorphic Poisson structures and compact holomorphic Poissonsubmanifolds 305. Stability of compact holomorphic Poisson submanifolds 46Appendix A. Deformations of Poisson structures 50Appendix B. Deformations of Poisson closed subschemes 52Appendix C. Simultaneous deformations of Poisson structures and Poisson closed subschemes 56References 601.
Introduction
In this paper, we study deformations of compact holomorphic Poisson submanifolds which extend Ko-daira’s series of papers on semi-regularity (deformations of compact complex submanifols of codimension 1)([KS59]), deformations of compact complex submanifolds of arbitrary codimensions ([Kod62]), and stabil-ity of compact complex submanifolds ([Kod63]) in the context of holomorphic Poisson deformations. Wewill review deformation theory of compact complex submainfolds presented in [KS59],[Kod62],[Kod63], andexplain how the theory can be extended in the context of holomorphic Poisson deformations.Let us review deformations of compact complex submanifolds of codimension 1 of a complex manifoldpresented in [KS59] where Kodaira-Spencer proved the theorem of completeness of characteristic systems ofcomplete continuous systems of semi-regular complex submanifolds of codimension 1. For the precise state-ment, we recall the definitions of a complex analytic family of compact complex submanifolds of codimension1, and maximality (or completeness) of a complex analytic family:
Definition 1.0.1.
Let W be a complex manifold of dimension n + 1 . We denote a point in W by w anda local coordinate of w by ( w , ..., w n +1 ) . By a complex analytic family of compact complex submanifolds ofcodimension of W , we mean a complex submanifold V ⊂ W × M of codimension where M is a complexmanifold, such that V t × t := ω − ( t ) = V ∩ π − ( t ) for each point t ∈ M is a connected compact complexsubmanifold of W × t , where ω : V → M is the map induced from the canonical projection π : W × M → M , The author was partially supported by NRF grant 2011-0027969. In this paper, all manifolds under consideration are paracompact and connected. and for each point p ∈ V , there is a holomorphic function S ( w, t ) on a neighborhood U p of p in W × M suchthat P n +1 α =1 | ∂S ( w,t ) ∂w α | = 0 at each point in U p ∩ V , and U p ∩ V is defined by S ( w, t ) = 0 . Definition 1.0.2.
Let
V ⊂ W × M ω −→ M be a complex analytic family of compact complex submanifoldsof W of codimension and let t be a point on M . We say that V ω −→ M is maximal at t if, for anycomplex analytic family V ′ ⊂ W × M ′ ω ′ −→ M ′ of compact complex submanifolds of W of codimension such that ω − ( t ) = ω ′− ( t ′ ) , t ′ ∈ M ′ , there exists a holomorphic map h of a neighborhood N ′ of t ′ on M ′ into M which maps t ′ to t such that ω ′− ( t ′ ) = ω − ( h ( t ′ )) for t ′ ∈ N ′ . We note that if we set aholomorphic map ˆ h : W × N ′ → W × M defined by ( w, t ′ ) → ( w, h ( t ′ )) , then the restriction map of ˆ h to V ′ | N ′ = ω ′− ( N ′ ) ⊂ W × N ′ defines a holomorphic map V ′ | N ′ → V so that V ′ | N ′ is the family induced from V by h , which means V ω −→ M is complete at t . Given a complex analytic family
V ⊂ W × M ω −→ M of compact complex submanifolds of codimension1, each fibre V t = ω − ( t ) of V for t ∈ M defines a complex line bundle N t on W . Then infinitesimaldeformations of V t in the family V are encoded in the cohomology group H ( V t , N t | V t ), and we can definethe characteristic map (see [KS59] p.479-480) ρ d,t : T t M → H ( V t , N t | V t )In [KS59], Kodaira-Spencer defined a concept of semi-regularity, and showed that the semi-regularity is ‘theright condition’ for ‘theorem of existence’ and thus ‘theorem of completeness’ for deformations of compactcomplex submanifolds of codimension 1 as follows. Definition 1.0.3.
Let V be a compact complex submanifold of W of codimension and N be the com-plex line bundle over W determined by V . Let r : N → N | V be the restriction map which induces ahomomorphism r ∗ : H ( W, N ) → H ( V , N | V ) . We say that V is semi-regular if r ∗ H ( W, N ) is zero. Theorem 1.0.4 (theorem of existence) . If V is semi-regular, then there exists a complex analytic family V ⊂ W × M ω −→ M of compact complex submanifolds of W containing V as the fibre ω − (0) over ∈ M such that the characteristic map ρ d, : T M → H ( V , N | V ) is an isomorphism. Theorem 1.0.5 (theorem of completeness) . Let
V ⊂ W × M ω −→ M be a complex analytic family of compactcomplex submanifolds of W of codimension . If the characteristic map ρ d, : T M → H ( V , N | V ) is an isomorphism, then the family V ω −→ M is maximal at the point t = 0 . In section 2, we extend the concept of semi-regularity and prove an analogue of theorem of existence(Theorem 1.0.4) and an analogue of theorem of completeness (Theorem 1.0.5) in the context of holomor-phic Poisson deformations. A holomorphic Poisson manifold W is a complex manifold whose structuresheaf is a sheaf of Poisson algebras. A holomorphic Poisson structure on W is encoded in a holomorphicsection (a holomorphic bivector field) Λ ∈ H ( W, ∧ T W ) with [Λ , Λ ] = 0, where T W is the sheaf ofgerms of holomorphic vector fields, and the bracket [ − , − ] is the Schouten bracket on W . In the sequela holomorphic Poisson manifold will be denoted by ( W, Λ ). Let V be a complex submanifold of a holo-morphic Poisson manifold ( W, Λ ) and let i : V ֒ → W be the embedding. Then V is called a holomorphicPoisson submanifold of ( W, Λ ) if V is a holomorphic Poisson manifold and the embedding i is a Poissonmap with respect to the holomorphic Poisson structures. Then the holomorphic Poisson structure on V is unique. Equivalently a holomorphic Poisson submanifold V of ( W, Λ ) can be characterized in the fol-lowing way: let V be covered by coordinate neighborhoods W i , i ∈ I in W . We choose a local coordinate( w i , z i ) := ( w i , ..., w ri , z i , ..., z di ) on each neighborhood W i such that w i = · · · = w ri = 0 defines V ∩ W i . Then V is a holomorphic Poisson submanifold of ( W, Λ ) if the restriction [Λ , w αi ] | V ∩ U i of [Λ , w αi ] to V ∩ U i is 0,i.e. [Λ , w αi ] | V ∩ U i := [Λ , w αi ] | w i =0 = 0 , α = 1 , ..., r , or [Λ , w αi ] is of the form: [Λ , w αi ] = P rβ =1 w βi T βiα ( w i , z i )for some T βiα ( w i , z i ) ∈ Γ( W i , T W ). We refer to [LGPV13] for general information on Poisson geometry
EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 3
Now we explain how we can extend the theory of deformations of compact complex submanifolds ofcodimension 1 to the theory of deformations of compact holomorphic Poisson submanifolds of a holomorphicPoisson manifold of codimension 1. We extend Definition 1.0.1 and Definition 1.0.2 to define concepts of afamily of compact holomorphic Poisson submanifolds of codimension 1 (see Definition 2.0.26), and maximality(or completeness) of a family of compact holomorphic Poisson submanifolds (see Definition 2.4.1).
Definition 1.0.6.
Let ( W, Λ ) be a holomorphic Poisson manifold of dimension n + 1 . We denote a point in W by w and a local coordinate of w by ( w , ..., w n +1 ) . By a Poisson analytic family of compact holomorphicPoisson submanifolds of codimension of ( W, Λ ) , we mean a holomorphic Poisson submanifold V ⊂ ( W × M, Λ ) of codimension where M is a complex manifold and Λ is the holomorphic Poisson structure on W × M induced from ( W, Λ ) , such that V t := ω − ( t ) = V ∩ π − ( t ) for each point t ∈ M is a connectedcompact holomorphic Poisson submanifold of ( W × t, Λ ) , where ω : V → M is the map induced from thecanonical projection π : W × M → M , and for each point p ∈ V , there is a holomorphic function S ( w, t ) on a neighborhood U p of p in W × M such that P n +1 α =1 | ∂S ( w,t ) ∂w α | = 0 at each point in U p ∩ V , and U p ∩ V isdefined by S ( w, t ) = 0 . Definition 1.0.7.
Let
V ⊂ ( W × M, Λ ) ω −→ M be a Poisson analytic family of compact holomorphic Poissonsubmanifolds of ( W, Λ ) of codimension and let t be a point on M . We say that V ω −→ M is maximal at t if, for any Poisson analytic family V ′ ⊂ ( W × M ′ , Λ ) ω ′ −→ M ′ of compact holomorphic Poisson submanifoldsof ( W, Λ ) of codimension such that ω − ( t ) = ω ′− ( t ′ ) , t ′ ∈ M ′ , there exists a holomorphic map h of aneighborhood N ′ of t ′ on M ′ into M which maps t ′ to t such that ω ′− ( t ′ ) = ω − ( h ( t ′ )) for t ′ ∈ N ′ . Wenote that if we set a Poisson map ˆ h : ( W × N ′ , Λ ) → ( W × M, Λ ) defined by ( w, t ′ ) → ( w, h ( t ′ )) , then therestriction map of ˆ h to V ′ | N ′ = ω ′− ( N ′ ) ⊂ ( W × N ′ , Λ ) defines a Poisson map V ′ | N ′ → V so that V ′ | N ′ isthe family induced from V by h , which means V ω −→ M is complete at t . Given a Poisson analytic family
V ⊂ ( W × M, Λ ) → M of compact holomorphic Poisson submanifoldsof codimension 1, each fibre V t = ω − ( t ) of V for t ∈ M defines a Poisson line bundle ( N t , ∇ t ) on ( W, Λ ),where ∇ t is the Poisson connection on N t which defines the Poisson line bundle structure (see [Kim14a]) sothat we have a complex of sheaves on W (see [Kim14a]) N • t : N t ∇ t −−→ N t ⊗ T W ∇ t −−→ N t ⊗ ∧ T W ∇ t −−→ · · · We will denote the i -th hypercohomology group by H i ( W, N • t ). We note that N • t induces, by restriction on V t , the complex of sheaves on V t N • t | V t : N t | V t ∇ t | Vt −−−−→ N t | V t ⊗ T W | V t ∇ t | Vt −−−−→ N t | V t ⊗ ∧ T W | V t ∇ t | Vt −−−−→ · · · We will denote the i -th hypercohomology group by H i ( V t , N • t | V t ). Then infinitesimal deformations of V t inthe family V are encoded in the cohomology group H ( V t , N • t | V t ), and we can define the characteristic map(see subsection 2.1) ρ d,t : T t M → H ( V t , N • t | V t )As in the concept of semi-regularity in [KS59], we similarly define a concept of Poisson semi-regularity andshow that Poisson semi-regularity (see Definition 2.1.8) implies ‘theorem of existence’ (see Theorem 2.2.3)and thus ‘theorem of completeness’ (see Theorem 2.4.2) for deformations of compact holomorphic Poissonsubmanifolds of codimension 1 as follows. Definition 1.0.8.
Let V be a compact holomorphic Poisson submanifold of a holomorphic Poisson manifold ( W, Λ ) of codimension and let ( N , ∇ ) be the Poisson line bundle over ( W, Λ ) determined by V . Wedenote by r : N • → N • | V the restriction map of the following complex of sheaves N • : N ∇ −−−−→ N ⊗ T W ∇ −−−−→ N ⊗ ∧ T W ∇ −−−−→ · · · y y y N • | V : N | V ∇ | V −−−−→ N | V ⊗ T W | V ∇ | V −−−−→ N | V ⊗ ∧ T W | V ∇ | V −−−−→ · · · which induces a homomorphism r ∗ : H ( W, N • ) → H ( V , N • | V ) . We say that V is Poisson semi-regularif the image r ∗ H ( W, N • ) is zero. CHUNGHOON KIM
Theorem 1.0.9 (Theorem of existence) . If V is Poisson semi-regular, then there exists a Poisson analyticfamily V ⊂ ( W × M, Λ ) ω −→ M of compact holomorphic Poisson submanifolds of ( W, Λ ) containing V asthe fibre ω − (0) over ∈ M such that the characteristic map ρ d, : T M → H ( V , N • | V ) ia an isomorphism. Theorem 1.0.10 (Theorem of completeness) . Let
V ⊂ ( W × M, Λ ) ω −→ M be a Poisson analytic family ofcompact Poisson submanifolds of ( W, Λ ) of codimension . If the characteristic map ρ d, : T M → H ( V , N • | V ) is an isomorphism, then the family V ω −→ M is maximal at the point t = 0 . Next we review deformation theory of compact complex submanifolds of arbitrary codimensions presentedin [Kod62] and explain how the theory can be extended to the theory of compact holomorphic Poissonsubmanifolds of arbitrary codimensions in terms of holomorphic Poisson deformations. In [Kod62], Kodairashowed that deformations of a compact complex submanifold V of a complex manifold W is controlledby the normal bundle N V/W of V in W so that infinitesimal deformations are encoded in the cohomologygroup H ( V, N V/W ) and obstructions are encoded in the cohomology group H ( V, N V/W ). For the precisestatement, we recall the following definition which generalize Definition 1.0.1 to arbitrary codimensions.
Definition 1.0.11 ([Kod62]) . Let W be a complex manifold of dimension d + r . We denote a point in W by w and a local coordinate of w by ( w , ..., w r + d ) . By a complex analytic family of compact complexsubmanifolds of dimension d of W , we mean a complex submanifold V ⊂ W × M of codimension r , where M is a complex manifold, such that (1) for each point t ∈ M , V t × t := ω − ( t ) = V ∩ π − ( t ) is a connected compact complex submanifoldof W × t of dimension d , where ω : V → M is the map induced from the canonical projection π : W × M → M . (2) for each point p ∈ V , there exist r holomorphic functions f α ( w, t ) , α = 1 , ..., r defined on a neighbor-hood U p of p in W × M such that rank ∂ ( f ,...,f r ) ∂ ( w ,...w r + d ) = r , and U p ∩ V is defined by the simultaneousequations f α ( w, t ) = 0 , α = 1 , ..., r .We call V ⊂ W × M a complex analytic family of compact complex submanifolds V t , t ∈ M of W . We alsocall V ⊂ W × M a complex analytic family of deformations of a compact complex submanifold V t of W foreach fixed point t ∈ M . We can define the concept of maximality (or completeness) of a complex analytic family of compactcomplex submanifolds of arbitrary codimenions as in Definition 1.0.2. Given a complex analytic family
V ⊂ W × M ω −→ M , for each fibre V t = ω − ( t ) of V for t ∈ M , infinitesimal deformations of V t in the family V are encoded in the cohomology group H ( V t , N V t /W ), and we can define the characteristic map (see [Kod62]p.147-150) σ t : T t M → H ( V t , N V t /W )In [Kod62], Kodaira showed that given a compact complex submanifold V of a complex manifold W , ob-structions of deformations of V in W are encoded in H ( V, N V/W ) and so when obstructions vanish, we canprove ‘theorem of existence’ and ‘theorem of completeness’ as follows.
Theorem 1.0.12 (theorem of existence) . Let V be a compact complex submanifold of a complex manifold W . If H ( V, N V/W ) = 0 , then there exists a complex analytic family V of compact complex submanifolds V t , t ∈ M of W such that V = V and the characteristic map σ : T ( M ) → H ( V, N V/W ) is an isomorphism. Theorem 1.0.13 (theorem of completeness) . Let V be a complex analytic family of compact holomorphicPoisson submanifolds V t , t ∈ M , of W . If the characteristic map σ : T ( M ) → H ( V , N V /W ) is an isomorphism, then the family V is maximal at t = 0 . EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 5
In section 3, we extend Definition 1.0.11 to define a concept of compact holomorphic Poisson submanifoldsof arbitrary codimensions, and prove an analogue of theorem of existence (Theorem 1.0.12) and an analogueof theorem of completeness (Theorem 1.0.13) in the context of holomorphic Poisson deformations. Let V be a holomorphic Poisson submanifold of a holomorphic Poisson manifold ( W, Λ ). Then we can define acomplex of sheaves associated with the normal bundle N V/W (see subsection 3.1 and Definition 3.1.13 ) N • V/W : N V/W ∇ −→ N V/W ⊗ T W | V ∇ −→ N V/W ⊗ ∧ T W | V ∇ −→ N V/W ⊗ ∧ T W | V ∇ −→ · · · which is called the complex associated with the normal bundle N V/W of a holomorphic Poisson submanifold V of a holomorphic Poisson manifold ( W, Λ ), and denote its i -th hypercohomology group by H i ( V, N • V/W ).Then holomorphic Poisson deformations of V in ( W, Λ ) is controlled by the complex of sheaves N • V/W so that infinitesimal deformations are encoded in the cohomology group H ( V, N • V/W ) and obstructionsare encoded in the cohomology group H ( V, N • V/W ). For the precise statement, we define a concept ofcompact holomorphic Poisson submanifolds of an arbitrary codimension which generalize Definition 1.0.6and Definition 1.0.11 as follows.
Definition 1.0.14 (compare Definition 1.0.11) . Let ( W, Λ ) be a holomorphic Poisson manifold of dimension d + r . We denote a point in W by w and a local coordinate of w by ( w , ..., w r + d ) . By a Poisson analyticfamily of compact holomorphic Poisson submanifolds of dimension d of ( W, Λ ) , we mean a holomorphicPoisson submanifold V ⊂ ( W × M, Λ ) of codimension r , where M is a complex manifold and Λ is theholomorphic Poisson structure on W × M induced from ( W, Λ ) , such that (1) for each point t ∈ M , V t × t := ω − ( t ) = V ∩ π − ( t ) is a connected compact holomorphic Poissonsubmanifold of ( W × t, Λ ) of dimension d , where ω : V → M is the map induced from the canonicalprojection π : W × M → M . (2) for each point p ∈ V , there exist r holomorphic functions f α ( w, t ) , α = 1 , ..., r defined on a neighbor-hood U p of p in W × M such that rank ∂ ( f ,...,f r ) ∂ ( w ,...w r + d ) = r , and U p ∩ V is defined by the simultaneousequations f α ( w, t ) = 0 , α = 1 , ..., r .We call V ⊂ ( W × M, Λ ) a Poisson analytic family of compact holomorphic Poisson submanifolds V t , t ∈ M of ( W, Λ ) . We also call V ⊂ ( W × M, Λ ) a Poisson analytic family of deformations of a compact holomorphicPoisson submanifold V t of ( W, Λ ) for each fixed point t ∈ M . We can define the concept of maximality (or completeness) of a Poisson analytic family of compactholomorphic Poisson submanifolds of arbitrary codimensions as in Definition 1.0.7. Given a Poisson analyticfamily
V ⊂ ( W × M, Λ ) ω −→ M of compact holomorphic Poisson submanifolds, for each fibre V t = ω − ( t )of V for t ∈ M , infinitesimal deformations of V t in the family V are encoded in the cohomology group H ( V t , N • V t /W ), and we can define the characteristic map (see subsection 3.2) σ t : T t M → H ( V t , N • V t /W )Given a compact holomorphic Poisson submanifold V of a holomorphic Poisson manifold ( W, Λ ), obstruc-tions of holomorphic Poisson deformations of V in ( W, Λ ) are encoded in H ( V, N • V/W ) and so when ob-structions vanish, we can prove ‘theorem of existence’ (see Theorem 3.3.1) and ‘theorem of completeness’(see Theorem 3.5.1) as fallows.
Theorem 1.0.15 (theorem of existence) . Let V be a compact holomorphic Poisson submanifold of a holo-morphic Poisson manifold ( W, Λ ) . If H ( V, N • V/W ) = 0 , then there exists a Poisson analytic family V ofcompact holomorphic Poisson submanifolds V t , t ∈ M of ( W, Λ ) such that V = V and the characteristicmap σ : T ( M ) → H ( V, N • V/W ) is an isomorphism. Theorem 1.0.16 (theorem of completeness) . Let V be a Poisson analytic family of compact holomorphicPoisson submanifolds V t , t ∈ M , of ( W, Λ ) . If the characteristic map σ : T ( M ) → H ( V , N • V /W ) is an isomorphism, then the family V is maximal at t = 0 . CHUNGHOON KIM
In section 4, we study simultaneous deformations of holomorphic Poisson structures and compact holomor-phic Poisson submanifolds. Let V be a holomorphic Poisson submanifold of a holomorphic Poisson manifold( W, Λ ). As we saw as above, the complex associated with the normal bundle N • V/W : N V/W ∇ −→ N V/W ⊗ T W | V ∇ −→ N V/W ⊗ ∧ T W | V ∇ −→ · · · (1.0.17)controls holomorphic Poisson deformations of V of ( W, Λ ), and the complex ∧ T • W : ∧ T W − [ − , Λ ] −−−−−→ ∧ T W − [ − , Λ ] −−−−−→ · · · (1.0.18)controls deformations of the holomorphic Poisson structure Λ on the fixed underlying complex manifold W (see Appendix A). By combining two complexes (1 . .
17) and (1 . . W (see subsection 4.1 and Definition 4.1.8)( ∧ T W ⊕ i ∗ N V/W ) • : ∧ T W ⊕ i ∗ N V/W ˜ ∇ −→ ∧ T W ⊕ i ∗ ( N V/W ⊗ T W | V ) ˜ ∇ −→ ∧ T W ⊕ i ∗ ( N V/W ⊗ ∧ T W | V ) ˜ ∇ −→ · · · where i : V ֒ → W is the embedding, which is called the extended complex associated with the normalbundle N V/W of a holomorphic Poisson submanifold V of a holomorphic Poisson manifold W , and denote its i -th hypercohomology group by H i ( V, ( ∧ T W ⊕ i ∗ N V/W ) • ). Then ( ∧ T W ⊕ i ∗ N V/W ) • controls simultaneousdeformations of Λ and V in ( W, Λ ) so that infinitesimal deformations are encoded in the cohomology group H ( W, ( ∧ T W ⊕ i ∗ N V/W ) • ) and obstructions are encoded in H ( W, ( ∧ T W ⊕ i ∗ N V/W ) • ). For the precisestatements, we extend Definition 1.0.14 of a Poisson analytic family of compact holomorphic submanifoldsof a holomorphic Poisson manifold ( W, Λ ) by deforming Λ on the fixed complex manifold W as well as aholomorphic Poisson submanifold V of ( W, Λ ) as follows (see Definition 4.0.1). Definition 1.0.19.
Let W be a complex manifold of dimension d + r . We denote a point in W by w anda local coordinate of w by ( w , ..., w r + d ) . By an extended Poisson analytic family of compact holomorphicPoisson submanifolds of dimension d of W , we mean a holomorphic Poisson submanifold V ⊂ ( W × M, Λ) of codimension r , where M is a complex manifold and Λ is a holomorphic Poisson structure on W × M ,such that (1) the canonical projection π : ( W × M, Λ) → M is a Poisson analytic family in the sense of [Kim14b]( but we allow non-compact fibres ) so that Λ ∈ H ( W × M, ∧ T W × M/M ) and π − ( t ) := ( W × t, Λ t ) is a holomorphic Poisson subsmanifold of ( W × M, Λ) for each point t ∈ M . (2) for each point t ∈ M , V t × t := ω − ( t ) = V ∩ π − ( t ) is a connected compact holomorphic Poissonsubmanifold of ( W × t, Λ t ) of dimension d , where ω : V → M is the map induced from π . (3) for each point p ∈ V , there exist r holomorphic functions f α ( w, t ) , α = 1 , ..., r defined on a neighbor-hood U p of p in W × M such that rank ∂ ( f ,...,f r ) ∂ ( w ,...w r + d ) = r , and U p ∩ V is defined by the simultaneousequations f α ( w, t ) = 0 , α = 1 , ..., r .We call V ⊂ ( W × M, Λ) an extended Poisson analytic family of compact holomorphic Poisson submanifolds V t , t ∈ M of ( W, Λ t ) . We also call V ⊂ ( W × M, Λ) an extended Poisson analytic family of simultaneousdeformations of a holomorphic Poisson submanifold V t of ( W, Λ t ) for each fixed point t ∈ M . As in Definition 1.0.2, we can similarly define the concept of maximality (or completeness) of an extendedPoisson analytic family (see Definition 4.5.1). Given an extended Poisson analytic family
V ⊂ ( M × M, Λ) ω −→ M of compact holomorphic Poisson submanifolds, for each fibre V t = ω − ( t ) ⊂ ( W, Λ t ) of V for t ∈ M ,infinitesimal deformations of (Λ t , V t ) in the family V are encoded in the cohomology group H ( W, ( ∧ T W ⊕ i ∗ N V t /W ) • ), and we can define the characteristic map (see subsection 4.2) σ t : T t M → H ( W, ( ∧ T W ⊕ i ∗ N V t /W ) • )Given a compact holomorphic Poisson manifold V of a compact holomorphic Poisson manifold ( W, Λ ),obstructions of simultaneous deformations of Λ and V are encoded in H ( W, ( ∧ T W ⊕ i ∗ N V/W ) • ) andso when obstructions vanish, we can prove ‘theorem of existence’ (see Theorem 4.3.1) and ‘theorem ofcompleteness’ (see Theorem 4.5.2) as follows. Theorem 1.0.20 (theorem of existence) . Let V be a holomorphic Poisson submanifold of a compact holo-morphic Poisson manifold ( W, Λ ) . If H ( W, ( ∧ T W ⊕ i ∗ N W/V ) • ) = 0 , then there exists an extended Poisson EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 7 analytic family
V ⊂ ( W × M , Λ) of compact holomorphic Poisson submanifolds V t , t ∈ M , of ( W, Λ t ) suchthat V = V ⊂ ( W, Λ ) and the characteristic map σ : T ( M ) → H ( W, ( ∧ T W ⊕ i ∗ N V/W ) • ) is an isomorphism. Theorem 1.0.21 (theorem of completeness) . Let
V ⊂ ( W × M , Λ) be an extended Poisson analytic familyof compact holomorphic Poisson submanifolds V t of ( W, Λ t ) . If the characteristic map ρ : T ( M ) → H ( W, ( ∧ T W ⊕ i ∗ N W/V ) • ) is an isomorphism, then the family V is maximal at t = 0 . Lastly we review stability of compact complex submanifolds presented in [Kod63] and explain how wecan extend the concept of stability in the context of holomorphic Poisson deformations. In [Kod63], Kodairadefined a concept of stability of compact complex submanifolds as follow:
Definition 1.0.22.
Let V be a compact complex submanifold of a complex manifold W . We call V a stablecomplex submanifold of W if and only if, for any complex fibre manifold ( W , B, p ) with p : W → B suchthat p − (0) = W for a point ∈ B , there exist a neighborhood N of in B and a complex fibre submanifold V with compact fibres of the complex fibre manifold W| N such that V ∩ W = V , where W| N is the restriction ( p − ( N ) , N, p ) of ( W , B, p ) to N . and he proved Theorem 1.0.23.
Let V be a compact complex submanifold of a complex manifold W . If the first cohomologygroup H ( V, N V/W ) vanishes, then V is a stable complex submanifold of W . In section 5, we similarly define a concept of stable compact holomorphic Poisson submanifolds as follows(see Definition 5.0.2):
Definition 1.0.24.
Let V be a compact holomorphic Poisson submanifold of a holomorphic Poisson manifold ( W, Λ ) . We call V a stable holomorphic Poisson submanifold of ( W, Λ ) if and only if, for any holomorphicPoisson fibre manifold ( W , Λ , B, p ) ( see Definition . . such that p − (0) = ( W, Λ ) for a point ∈ B ,there exist a neighborhood N of in B and a holomorphic Poisson fibre submanifold V with compact fibresof the holomorphic Poisson fibre manifold ( W , Λ) | N such that V ∩ W = V , where ( W , Λ) | N is the restriction ( p − ( N ) , Λ | p − ( N ) , N, p ) of ( W , Λ , B, p ) to N . and we prove (see Theorem 5.1.1) Theorem 1.0.25.
Let V be a compact holomorphic Poisson submanifold of a holomorphic Poisson manifold ( W, Λ ) . If the first cohomology group H ( V, N • V/W ) vanishes, then V is a stable holomorphic Poissonsubmanifold of ( W, Λ ) . In appendices B and C, we present deformations of Poisson closed subschemes in the language of functorsof Artin rings which is the algebraic version of deformations of holomorphic Poisson submanifolds. Weidentity first-order deformations and obstructions (see Proposition B.2.4 and Proposition C.2.2).2.
Deformations of compact holomorphic Poisson submanifolds of codimension andPoisson semi-regularity Definition 2.0.26.
Let ( W, Λ ) be a holomorphic Poisson manifold. We denote a point in W by w anda local coordinate of w by ( w , ..., w n +1 ) . By a Poisson analytic family of compact holomorphic Poissonsubmanifolds of codimension of ( W, Λ ) , we mean a holomorphic Poisson submanifold V ⊂ ( W × M, Λ ) of codimension where M is a complex manifold and Λ is the holomorphic Poisson structure on W × M induced from ( W, Λ ) , such that V t × t := ω − ( t ) = V ∩ π − ( t ) for each point t ∈ M is a connected compactholomorphic Poisson submanifold of ( W × t, Λ ) , where ω : V → M is the map induced from the canonicalprojection π : W × M → M , and for each point p ∈ V , there is a holomorphic function S ( w, t ) on aneighborhood U p of p in W × M such that P n +1 α =1 | ∂S ( w,t ) ∂w α | = 0 at each point in U p ∩ V , and U p ∩ V is definedby S ( w, t ) = 0 . for the definition of a complex fibre manifold and a complex fibre submanifold, see [Kod63] p.79 CHUNGHOON KIM
Infinitesimal deformations.
Let ( W, Λ ) be a holomorphic Poisson manifold. We denote by w a point on ( W, Λ ) and by ( w , ..., w n +1 )the local holomorphic coordinates (not specified) of w . Consider a small spherical neighborhood N of a pointon M and let { U i } be a locally finite covering of W by sufficiently small coordinate neighborhoods U i .Consider a Poisson analytic family V ⊂ ( W × M, Λ ) of compact holomorphic Poisson submanifolds ofcodimension 1 as in Definition 2.0.26. Let N be a small spherical neighborhood of a point on M . Then theholomorphic Poisson submanifold V of ( W × M, Λ ) is defined in each neighborhood U i × N by a holomorphicequation S i ( w, t ) = 0 where S i ( w, t ) is a holomorphic function on U i × N such that P n +1 α =1 (cid:12)(cid:12)(cid:12) ∂S i ( w,t ) ∂w α (cid:12)(cid:12)(cid:12) = 0 ateach point ( w, t ) of V ∩ ( U i × N ) (if V ∩ ( U i × N ) is empty, we set S i ( w, t ) = 1). By letting S i ( w, t ) = f ik ( w, t ) S k ( w, t ) , w ∈ U i ∩ U k , (2.1.1)we get a system { f ik ( w, t ) } of non-vanishing holomorphic functions f ik ( w, t ) defined, respectively, on ( U i × N ) ∩ ( U k × N ) satisfying f ik ( w, t ) = f ij ( w, t ) f jk ( w, t ) , w ∈ U i ∩ U j ∩ U k . (2.1.2)On the other hand, since S i ( w, t ) = 0 define a holomorphic Poisson submanifold, [Λ , S i ( w, t )] is of the form[Λ , S i ( w i , t )] = S i ( w, t ) T i ( w, t )(2.1.3)for some T i ( w, t ) = P n +1 i =1 T αi ( w, t ) ∂∂w α , where T αi ( w, t ) is a holomorphic function on U i × N . By taking[Λ , − ] on (2 . . − [Λ , S i ( w, t )] ∧ T i ( w, t )]+ S i ( w, t )[Λ , T i ( w, t )] = − S i ( w, t ) T i ( w, t ) ∧ T i ( w, t )+ S i ( w, t )[Λ , T i ( w, t )] = S i ( w, t )[Λ , T i ( w, t )] so that[Λ , T i ( w, t )] = 0 , w ∈ U i (2.1.4)From (2 . .
1) and (2 . . f ik ( w, t ) S k ( w, t ) T i ( w, t ) = S i ( w, t ) T i ( w, t ) = [Λ , S i ( w, t )] = [Λ , f ik ( w, t ) S k ( w, t )] = S k ( w, t )[Λ , f ik ( w, t )] + f ik ( w, t )[Λ , S k ( w, t )] = S k ( w, t )[Λ , f ik ( w, t )] + f ik ( w, t ) S k ( w, t ) T k ( w, t ) so that f ik ( w, t ) T i ( w, t ) − f ik ( w, t ) T k ( w, t ) = [Λ , f ik ( w, t )] , w ∈ U i ∩ U k (2.1.5)Then the conditions (2.1.2),(2.1.4),(2.1.5) show that ( { f ik ( w, t ) } , { T i ( w, t ) } ) defines the Poisson line bundleover ( W, Λ ) for each t ∈ N . We will denote the Poisson line bundle by ( N t , ∇ t ) for t ∈ N , where ∇ t is thePoisson connection on N t which defines the Poisson line bundle structure (see [Kim14a]). Then we have acomplex of sheaves on W (see [Kim14a]) N • t : N t ∇ t −−→ N t ⊗ T W ∇ t −−→ N t ⊗ ∧ T W ∇ t −−→ · · · We will denote the i -th hypercohomology group by H i ( W, N • t ). We note that N • t induces, by restriction on V t , the complex of sheaves on V t N • t | V t : N t | V t ∇ t | Vt −−−−→ N t | V t ⊗ T W | V t ∇ t | Vt −−−−→ N t | V t ⊗ ∧ T W | V t ∇ t | Vt −−−−→ · · · We will denote the i -th hypercohomology group by H i ( V t , N • t | V t ).Denote by ( t , .., t m ) a system of holomorphic coordinates on N . For any tangent vector v = P mr =1 v r ∂∂t r of M at t ∈ N , we set ψ i ( w, t ) = − P mr =1 v r ∂S i ( w,t ) ∂t r . Since S i ( w, t ) = 0 for w ∈ V t , by taking the derivativeof (2 . .
1) and (2 . .
3) with respect to t , we obtain ψ i ( w, t ) = f ik ( w, t ) ψ k ( w, t ) , w ∈ V t ∩ U i ∩ U k , (2.1.6) [Λ , ψ i ( w, t )] | V t =[Λ , ψ i ( w, t )] | S i ( w,t )=0 = ψ i ( w, t ) T i ( w, t ) , w ∈ V t ∩ U i (2.1.7)From (2 . .
6) and (2 . . { ψ i ( w, t ) } define an element in H ( V t , N • t | V t ) so that we have a linear map ρ d,t : T t M → H ( V t , N • t | V t ) ∂∂t ∂V t ∂t := { ψ i ( z i , t ) } We call ρ d,t the characteristic map. Definition 2.1.8.
Let V be a compact holomorphic Poisson submanifold of a holomorphic Poisson manifold ( W, Λ ) of codimension and let ( N , ∇ ) be the Poisson line bundle over ( W, Λ ) determined by V . Wedenote by r : N • → N • | V the restriction map of the following complex of sheaves EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 9 N • : N ∇ −−−−→ N ⊗ T W ∇ −−−−→ N ⊗ ∧ T W ∇ −−−−→ · · · y y y N • | V : N | V ∇ | V −−−−→ N | V ⊗ T W | V ∇ | V −−−−→ N | V ⊗ ∧ T W | V ∇ | V −−−−→ · · · which induces a homomorphism r ∗ : H ( W, N • ) → H ( V , N • | V ) . We say that V is Poisson semi-regularif the image r ∗ H ( W, N • ) is zero. Theorem of existence.
We extend the argument in [KS59] in the context of Poisson deformations (see [KS59] p.484-493). Wetried to maintain notational consistency with [KS59]. Let ( W, Λ ) be a holomorphic Poisson manifold ofdimension n + 1 ≥ V be a compact holomorphic Poisson submanifold of ( W, Λ ) of dimension n . Inwhat follows we denote by p a point on W and by ( w ( p ) , , , , , w n +1 ( p )) the coordinate of p with respect toa system of local holomorphic coordinate ( w , ..., w n +1 ) on W . We choose a locally finite covering U = { U i } of W such that(1) each neighborhood U i is a polycylinder: U i = { p || w i ( p ) | < , ..., | w n +1 i ( p ) | < } where ( w i , ..., w n +1 i )is a system of local holomorphic coordinates which covers the closure of U i .(2) V ∩ U i coincides with the coordinate plane w n +1 i = 0 if V ∩ U i is not empty,(3) V ∩ U i ∩ U k is not empty if V ∩ U i , V ∩ U k and U i ∩ U k are not empty.(4) if V ∩ U i is not empty, we write w i = w n +1 i , z i = w i , ..., z ni = w ni .(5) { U δi } covers W , where U δi = { p || w i ( p ) | < − δ, ..., | z i ( p ) | < − δ, ..., | z ni ( p ) | < − δ, | w i ( p ) | < } fora sufficiently small δ .Let S i | ( p ) = w i ( p ) if V ∩ U i = ∅ and S i | ( p ) = 1 if V ∩ U i = ∅ for p ∈ U i . If we let f ik | ( p ) := S | ( p ) S k | ( p ) , for p ∈ U i ∩ U k ,T i | ( p ) := [Λ , log S i | ( p )]( i.e [Λ , S i | ( p )] = S i | ( p ) T i | ( p )) , for p ∈ U i , then the system ( { f ik | ( p ) } , { T i | ( p ) } ) defines the Poisson line bundle N . Let { β , ..., β m } be a basis of H ( V , N • | V ). Each β r is a holomorphic section of N over V and β r is written in the form via theidentification N | U i ∼ = U i × C , β r : p ( p, β ri ( p ))where β ri ( p ) are holomorphic functions on V ∩ U i satisfying β ri ( p ) = f ik | ( p ) β rk ( p ) , for p ∈ V ∩ U i ∩ U k .(2.2.1) − [Λ , β ri ( p )] | V + β ri ( p ) T i | ( p ) = 0 , for p ∈ V ∩ U i (2.2.2)With this preparation, we prove Theorem 2.2.3 (Theorem of existence) . If V is Poisson semi-regular, then there exists a Poisson analyticfamily V ⊂ ( W × M, Λ ) ω −→ M of compact holomorphic Poisson submanifolds of ( W, Λ ) containing V asthe fibre ω − (0) over ∈ M such that the characteristic map ρ d, : T M → H ( V , N | • V ) ∂∂t (cid:18) ∂V t ∂t (cid:19) t =0 is an isomorphism.Proof. Let N be a spherical neighborhood of 0 on the space of m complex variables t , ..., t m , where m =dim H ( V , N • | V ). In order to prove Theorem 2.2.3, it suffices to construct a system { S i ( p, t ) } of holomorphicfunctions S i ( p, t ) defined on U i × N and a system { f ik ( p, t ) } of non-vanishing holomorphic functions f ik ( p, t )defined on ( U i ∩ U k ) × N and a system { T i ( p, t ) } of holomorphic vector fields T i ( p, t ) defined on U i × N such that S i ( p, t ) = f ik ( p, t ) S k ( p, t ) , for p ∈ U i ∩ U k , (2.2.4) [Λ , S i ( p, t )] = S i ( p, t ) T i ( p, t ) , for p ∈ U i , (2.2.5) S i ( p,
0) = S i | ( p ) , f ik ( p,
0) = f ik | ( p ) , T i ( p,
0) = T i | ( p )(2.2.6) S i ( p, t ) = 0 , if V ∩ U i = ∅ (2.2.7) ∂S i ( p, t ) ∂t r | t =0 = β ri ( p ) , r = 1 , ..., m. (2.2.8)We write S i ( p, t ) , f ik ( p, t ) and T i ( p, t ) in the forms S i ( p, t ) = S i | ( p ) + ∞ X µ =1 S i | µ ( p, t ) , f ik ( p, t ) = f ik | ( p ) + ∞ X µ =1 f ik | µ ( p, t ) , T i ( p, t ) = T i | ( p ) + ∞ X µ =1 T i | µ ( p, t )where S i | µ ( p, t ) , f ik | µ ( p, t ) and T i | µ ( p, t ) are homogenous polynomials in t = ( t , ..., t m ) of degree µ whosecoefficients are holomorphic functions on U i , on U i ∩ U k , and holomorphic vector fields on U i , respectively.Let S µi ( p, t ) = S i | ( p ) + µ X λ =1 S i | λ ( p, t )(2.2.9) f µik ( p, t ) = f ik | ( p ) + µ X λ =1 f ik | λ ( p, t )(2.2.10) T µi ( p, t ) = T i | ( p ) + µ X λ =1 T i | λ ( p, t )(2.2.11) Notation 1.
We write the power series expansion of a holomorphic function P ( t ) in t , ..., t m defined on aneighborhood of the origin in the form : P ( t ) = P ( t ) + P ( t ) + · · · + P µ ( t ) + · · · , where each P µ ( t ) denotesa homogenous polynomial of degree µ in t , ..., t m . We set P µ ( t ) := P (0) + P ( t ) + · · · + P µ ( t ) . We write [ P ( t )] µ for P µ ( t ) when we substitute a complicated expression for P ( t ) . For any power series P ( t ) , and Q ( t ) in t = ( t , ..., t m ) , we indicate P ( t ) ≡ µ Q ( t ) that P ( t ) − Q ( t ) contains no term of degree ≤ µ in t . By Notation 1, (2 . .
4) and (2 . .
5) are equivalent to the system of congruences S µi ( p, t ) ≡ µ f µik ( p, t ) S µk ( p, t )(2.2.12) [Λ , S µi ( p, t )] ≡ µ S µi ( p, t ) T µi ( p, t ) , µ = 1 , , , · · · (2.2.13)We will construct S i | µ ( p, t ) , f ik | µ ( p, t ) by induction on µ satisfying (2 . .
12) and (2 . . S i | µ ( p, t ) , µ ≥ S i | µ ( p, t ) = ( ψ i | µ ( z i ( p ) , t ) , if V ∩ U i = ∅ , if V ∩ U i = ∅ where z i ( p ) = ( z i ( p ) , ..., z ni ( p )) and ψ i | µ ( z i , t ) is a homogeneous polynomial of degree µ in t whose coefficientsare holomorphic functions of z i = ( z i , ..., z ni ) defined on the polycylinder: | z i | < , ..., | z ni | < ψ i | ( z i , t ) by ψ i | ( z i ( p ) , t ) = m X r =1 t r β ri ( p ) , p ∈ V ∩ U i . (2.2.15)and determine S i | ( p, t ) , S i ( p, t ) by (2 . .
14) and (2 . . f ik | ( p, t ) = S i | ( p,t ) − f ik | ( p ) S k | ( p,t ) S k | ( p ) , f ik ( p, t ) = f ik | ( p ) + f ik | ( p, t ) is holomorphic in p and satisfies (2 . . (for the detail, see [KS59] p.486).On the other hand, we note that [Λ , S i ( p, t )] ≡ S i ( p, t ) T i ( p, t ) is equivalent to [Λ , S i | ( p, t )] = S i | ( p, t ) T i | ( p ) + S i | ( p ) T i | ( p, t ). By letting T i | ( p, t ) = [Λ , S i | ( p, t )] − S i | ( p, t ) T i | ( p ) S i | ( p ) , EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 11 T i ( p, t ) = T i | ( p ) + T i | ( p, t ) is holomorphic in p by (2 . .
2) and satisfies (2 . . .Now suppose that we have already constructed S µi ( p, t ) , f µik ( p, t ) , T µi ( p, t ) satisfying (2 . . µ and (2 . . µ which imply that f µik ( p, t ) ≡ µ f µij ( p, t ) f µjk ( p, t )(2.2.16) [Λ , T µi ( p, t )] = 0(2.2.17) [Λ , f µik ( p, t )] + f µik ( p, t ) T µk ( p, t ) − f µik ( p, t ) T µi ( p, t ) ≡ µ ψ ik | µ +1 ( p, t ) in t of degree µ + 1 whose coefficients are in Γ( V ∩ U i ∩ U k , O V ) and homogenous polynomials W i | µ +1 ( p, t ) of degree µ + 1 whose coefficient are in Γ( U i ∩ V , T W | V )by ψ ik | µ +1 ( p, t ) ≡ µ +1 f µik ( p, t ) S µk ( p, t ) − S µi ( p, t ) for p ∈ V ∩ U i ∩ U k (2.2.19) W i | µ +1 ( p, t ) ≡ µ +1 [Λ , S µi ( p, t )] | V − S µi ( p, t ) T µi ( p, t ) , for p ∈ V ∩ U i (2.2.20)Then we have (for the detail, see [KS59] p.487) ψ ik | µ +1 ( p, t ) = ψ ij | µ +1 ( p, t ) + f ij | ( p ) ψ jk | µ +1 ( p, t ) , for p ∈ V ∩ U i ∩ U j ∩ U k (2.2.21)On the other hand, let ˜ W i | µ +1 ( p, t ) ≡ µ +1 [Λ , S µi ( p, t )] − S µi ( p, t ) T µi ( p, t ) for p ∈ U i . Then from (2.2.17),we obtain[Λ , ˜ W i | µ +1 ( p, t )] ≡ µ +1 − [Λ , S µi ( p, t ) T µi ( p, t )] ≡ µ +1 [Λ , S µi ( p, t )] ∧ T µi ( p, t ) − S µi ( p, t )[Λ , T µi ( p, t )] ≡ µ +1 ˜ W i | µ +1 ( p, t ) ∧ T µi ( p, t ) + S µi ( p, t ) T µi ( p, t ) ∧ T µi ( p, t ) − S µi ( p, t )[Λ , T µi ( p, t )] ≡ µ +1 ˜ W i | µ +1 ( p, t ) ∧ T i | ( p )By restricting to V , equivalently by taking S i | ( p ) = 0, we get − [ W i | µ +1 ( p, t ) , Λ ] | V + ( − W i | µ +1 ( p, t ) ∧ T i | ( p ) = 0(2.2.22)Lastly, let G ik | µ +1 ( p, t ) ≡ µ +1 f µik ( p, t )( T µi ( p, t ) − T µk ( p, t )) − [Λ , f µik ( p, t )]. We note that G ik | µ +1 ( p, t ) ≡ µ V , equivalently by taking S i | ( p ) = 0,[Λ , ψ ik | µ +1 ( p, t )] | V ≡ µ +1 [Λ , f µik ( p, t ) S µk ( p, t )] | V − [Λ , S µi ( p, t )] | V ≡ µ +1 f µik ( p, t )[Λ , S µk ( p, t )] | V + S µk ( p, t )[Λ , f µik ( p, t )] | V − [Λ , S µi ( p, t )] | V ≡ µ +1 f µik ( p, t ) W k | µ +1 ( p, t ) + f µik ( p, t ) S µk ( p, t ) T µk ( p, t ) + S µk ( p, t )[Λ , f µik ( p, t )] | V − W i | µ +1 ( p, t ) − S µi ( p, t ) T µi ( p, t ) ≡ µ +1 f µik ( p, t ) W k | µ +1 ( p, t ) + f µik ( p, t ) S µk ( p, t ) T µk ( p, t ) + S µk ( p, t ) f µik ( p, t )( T µi ( p, t ) − T µk ( p, t )) − S µk ( p, t ) G ik | µ +1 ( p, t ) − W i | µ +1 ( p, t ) − S µi ( p, t ) T µi ( p, t ) ≡ µ +1 f µik ( p, t ) W k | µ +1 ( p, t ) − W i | µ +1 ( p, t ) + ( f µik ( p, t ) S µk ( p, t ) − S µi ( p, t )) T µi ( p, t ) − S k | ( p ) G ik | µ +1 ( p, t ) ≡ µ +1 f ik | ( p ) W k | µ +1 ( p, t ) − W i | µ +1 ( p, t ) + ψ ik | µ +1 ( p, t ) T i | ( p )Hence we obtain − [ ψ ik | µ +1 ( p, t ) , Λ ] | V + ψ ik | µ +1 ( p, t ) T i | ( p ) + f ik | ( p ) W k | µ +1 ( p, t ) − W i | µ +1 ( p, t ) = 0(2.2.23)Hence from (2 . . . .
22) and (2 . . ψ µ +1 ( t ) , W µ +1 ( t )) := ( { ψ ik | µ +1 ( p, t ) } , { W i | µ +1 ( p, t ) } ) definesa 1-cocycle in the following ˇCech resolution of N • | V : C ( U ∩ V , N | V ⊗ ∧ T W | V ) ∇ | V x C ( U ∩ V , N | V ⊗ T W | V ) δ −−−−→ C ( U ∩ V , N | V ⊗ T W | V ) ∇ | V x ∇ | V x C ( U ∩ V , N | V ) − δ −−−−→ C ( U ∩ V , N | V ) δ −−−−→ C ( U ∩ V , N | V ) Next we show that if ( ψ µ +1 ( t ) , W µ +1 ( t )) vanishes identically, we can construct S µ +1 i ( p, t ) , f µ +1 ik ( p, t ) and T µ +1 i ( p, t ) satisfying (2 . . µ +1 and (2 . . µ +1 . Indeed, let us assume that ( ψ µ +1 ( t ) , W µ +1 ( t )) vanishesidentically. Then there exists homogenous polynomials φ i | µ +1 ( p, t ) in t of degree µ + 1 whose coefficients areholomorphic functions on V ∩ U i such that ψ ik | µ +1 ( p, t ) = φ i | µ +1 ( p, t ) − f ik | ( p ) φ k | µ +1 ( p, t ) , p ∈ V ∩ U i ∩ U k , (2.2.24) W i | µ +1 ( p, t ) = − [Λ , φ i | µ +1 ( p, t )] | V + φ i | µ +1 ( p, t ) T i | ( p ) , p ∈ V ∩ U i . (2.2.25)We define ψ i | µ +1 ( z i , t ) := ψ i | µ +1 ( z i ( p ) , t ) = φ i | µ +1 ( p, t ) , p ∈ V ∩ U i and determine S i | µ +1 ( p, t ) by (2.2.14),and S µ +1 i ( p, t ) by (2.2.9). We define f ik | µ +1 ( p, t ) ≡ µ +1 S µi ( p,t ) − f µik ( p,t ) S µk ( p,t )+ S i | µ +1 ( p,t ) − f ik | ( p ) S k | µ +1 ( p,t ) S k | ( p ) anddetermine f µ +1 ik ( p, t ) by (2.2.10). Then S µ +1 i ( p, t ) and f µ +1 ik ( p, t ) satisfy (2 . . µ +1 (for the detail, see [KS59]p.488).On the other hand, we note that from (2.2.25), we have W i | µ +1 ( p, t ) = − [Λ , S i | µ +1 ( p, t )] | V + S i | µ +1 ( p, t ) T i | ( p ) , p ∈ V ∩ U i . (2.2.26)We set T i | µ +1 ( p, t ) : ≡ µ +1 Φ µ ( p, t ) S i | ( p ) = [Λ , S µi ( p, t )] − S µi ( p, t ) T µi ( p, t ) + [Λ , S i | µ +1 ( p, t )] − S i | µ +1 ( p, t ) T i | ( p ) S i | ( p )(2.2.27)Then Φ µ ( p, t ) ≡ µ +1 p ∈ V ∩ U i from (2 . .
20) and (2 . .
26) so that T i | µ +1 ( p, t ) are holomorphic in p andwe have, from (2.2.27),[Λ , S µi ( p, t ) + S i | µ +1 ( p, t )] ≡ µ +1 S µi ( p, t ) T µi ( p, t ) + S i | µ +1 ( p, t ) T i | ( p ) + S i | ( p ) T i | µ +1 ( p, t ) ≡ µ +1 ( S µi ( p, t ) + S i | µ +1 ( p, t ))( T µi ( p, t ) + T i | µ +1 ( p, t ))so that S µ +1 i ( p, t ) and T µ +1 i ( p, t ) satisfy (2 . . µ +1 .Now we prove that ( ψ µ +1 ( t ) , W µ +1 ( t )) vanishes identically if V is Poisson semi-regular. For this purposesit suffices to construct a polynomial ( { η ki ( p, t ) } , { ω i ( p, t ) } ) in t of degree µ + 1 with coefficients in H ( W, N • )such that ψ ik | µ +1 ( p, t ) = η ik ( p, t ) for p ∈ V ∩ U i ∩ U k , (2.2.28) W i | µ +1 ( p, t ) = ω i ( p, t ) for p ∈ V ∩ U i . (2.2.29)In fact, { η ik ( p, t ) , ω i ( p, t ) } represents a polynomial ( η ( t ) , ω ( t )) in t with coefficients in H ( W, N • ), and(2 . .
28) and (2 . .
29) imply that ( ψ µ +1 ( t ) , W µ +1 ( t )) = r ∗ ( η ( t ) , ω ( t )). Hence we obtain ( ψ µ +1 ( t ) , W µ +1 ( t ))vanishes if V is Poisson semi-regular. Lemma 2.2.30 (see [KS59] Lemma 1 p.488) . For each integer λ ≤ µ , there exist polynomials g ik := g λik ( p, t ) in t of degree λ whose coefficients are holomorphic functions in p defined on U i ∩ U k such that f ik | ( p ) exp g λik ( p, t ) ≡ λ f λik ( p, t )(2.2.31)We define polynomials ˆ f µ +1 ik := ˆ f µ +1 ik ( p, t ) ≡ µ +1 f ik | ( p ) exp g µik ( p, t ) in t of degree µ +1. Then ˆ f µ +1 ik ( p, t ) ≡ µ f µik ( p, t ) and S µi ( p, t ) ≡ µ ˆ f µ +1 ( p, t ) S µk ( p, t ). By letting η ik ( p, t ) ≡ µ +1 ˆ f µ +1 ik ( p, t ) S µk ( p, t ) − S µi ( p, t ), we get (forthe detail, see [KS59] p.489) η ik ( p, t ) = η ij ( p, t ) + f ij | ( p ) η jk ( p, t )(2.2.32)On the other hand, writing f ik := f ik ( p, t ) and T i := T i ( p, t ), we note that from (2 . . , f µik ] + f µik ( T µk − T µi ) ≡ µ
0. Since f ik | exp( g µik ) ≡ µ f µik by (2.2.31), we have[Λ , f ik | exp g µik ] + f ik | exp( g µik )( T µk − T µi ) ≡ µ ⇐⇒ [Λ , log( f ik | exp g µik )] + ( T µk − T µi ) ≡ µ ⇐⇒ [Λ , log f ik | ] + [Λ , g µik ] + ( T µk − T µi ) = 0 ⇐⇒ [Λ , ˆ f µ +1 ik ] + ˆ f µ +1 ik ( T µk − T µi ) = 0(2.2.33) EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 13
From (2 . . , S µi ( p, t )] ≡ µ S µi ( p, t ) T µi ( p, t ). We define ω i := ω i ( p, t ) ≡ µ +1 [Λ , S µi ( p, t )] − S µi ( p, t ) T µi ( p, t ). Then we have from (2.2.33)[Λ , η ik ] ≡ µ +1 [Λ , ˆ f µ +1 ik S µk ] − [Λ , S µi ] ≡ µ +1 S µk [Λ , f µ +1 ik ] + ˆ f µ +1 ik [Λ , S µk ] − [Λ , S µi ] ≡ µ +1 S µk [Λ , ˆ f µ +1 ik ] + ˆ f µ +1 ik S µk T µk + f ik | ω k − S µi T µi − ω i ≡ µ +1 S µk [Λ , ˆ f µ +1 ik ] + ˆ f µ +1 ik S µk T µk + f ik | ω k + ( η ik − ˆ f µ +1 ik S µk ) T µi − ω i ≡ µ +1 S µk ([Λ , ˆ f µ +1 ik ] + ˆ f µ +1 ik T µk − ˆ f µ +1 ik T µi ) + η ik T i | + f ik | ω k − ω i ≡ µ +1 η ik T i | + f ik | ω k − ω i so that we obtain the equality − [ η ik ( p, t ) , Λ ] + η ik ( p, t ) T i | ( p ) + f ik | ( p ) ω k ( p, t ) − ω i ( p, t ) = 0 , p ∈ U i ∩ U k . (2.2.34)Lastly, we have from (2.2.17),[Λ , ω i ] ≡ µ +1 − [Λ , S µi T µi ] ≡ µ +1 [Λ , S µi ] ∧ T µi − S µi [Λ , T µi ] ≡ µ +1 ω i ∧ T µi + S µi T µi ∧ T µi − S µi [Λ , T µi ] ≡ µ +1 ω i ∧ T i | so that we obtain the equality − [ ω i ( p, t ) , Λ ] + ( − ω i ( p, t ) ∧ T i | ( p ) = 0 , p ∈ U i . (2.2.35)Form (2.2.32), (2.2.34), and (2.2.35), ( { η ik ( p, t ) } , { ω i ( p, t ) } ) is a polynomial in t whose coefficients in H ( W, N ). Then since S µk ( p, t ) ≡ p ∈ V ∩ U k , and ˆ f µ +1 ik ( p, t ) ≡ µ f µik ( p, t ), we obtain η ik ( p, t ) ≡ µ +1 f µik ( p, t ) S µk ( p, t ) − S µi ( p, t ) ≡ µ +1 ψ ik | µ +1 ( p, t ) , p ∈ V ∩ U k ,ω i ( p, t ) ≡ µ +1 [Λ , S µi ( p, t )] | V − S µi ( p, t ) T µi ( p, t ) ≡ µ +1 W i | µ +1 ( p, t ) , p ∈ V ∩ U i . Hence when V is Poisson semi-regular, we can construct S µi ( p, t ) , f µik ( p, t ) and T µi ( p, t ) satisfying (2 . . µ and (2 . . µ by induction on µ , and therefore we obtain formal power sereis S i ( p, t ) , f ik ( p, t ) and T i ( p, t )satisfying (2.2.4),(2.2.5),(2.2.6),(2.2.7) and (2.2.8).2.3. Proof of convergence.Notation 2.
Consider a formal power series f ( t ) = f ( p, t ) = P f h h ··· h m ( p )( t ) h ( t ) h · · · ( t m ) h m whosecoefficients f h h ··· h m ( p ) are vector-valued holomorphic functions in p defined on a domain and a power series a ( t ) = P a h h ··· h m ( t ) h ( t ) h · · · ( t m ) h m , a h h ··· h m ≥ . We indicate by f ( p, t ) ≪ a ( t ) that | f h ··· h m ( p ) | .
2. Here we write T i ( t ) = T i ( p, t ) by the form T i ( p, t ) = T i ( p, t ) ∂∂z i + · · · T ni ( p, t ) ∂∂z ni + T n +1 i ( p, t ) ∂∂w i by which we consider T i ( p, t ) a power series in t whose coefficients are vector-valued holomorphic functionson U i .We may assume that | f ik ( p ) | < c , p ∈ U i ∩ U k for some constant c >
0. Then S i ( t ) − S i | ≪ A ( t ) if b issufficiently large. Suppose that f µ − ik ( t ) − f ik | ≪ c A ( t ) , p ∈ U i ∩ U k (2.3.5) S µi ( t ) − S i | ≪ A ( t ) , p ∈ U i (2.3.6) T µ − i ( t ) − T i | ≪ d A ( t ) , p = ( z i ( p ) , w i ( p )) ∈ U i with | w i ( p ) | < , | z i ( p ) | < − δ ⇐⇒ p ∈ U δi (2.3.7)First we show that f µik ( t ) − f ik | ≪ c A ( t ) , p ∈ U i ∩ U k for some constant c >
0. We briefly summarizeKodaira’s result in the following (see [KS59] p.491-492): by setting c = c +2)( c + c ) ǫ for some sufficientlysmall constant 0 < ǫ <
1, and assuming c > bc (1 + c ) , (2.3.8)we get f µik ( t ) − f ik | ≪ c A ( t ) , p ∈ U i ∩ U k .Next we show that T µi ( t ) − T i | ≪ d A ( t ) , p ∈ U δi (2.3.9)for some constant d >
0. We may assume that | T i | ( p ) | < d , p ∈ U i (2.3.10)for some constant d >
0. We recall from (2.2.27) that T i | µ ( p, t ) ≡ µ [Λ , S µ − i ] − T µ − i S µ − i + [Λ , S i | µ ] − T i | S i | µ S i | ≡ µ [Λ , S µi ] − T µ − i S µi S i | (2.3.11)We estimate [Λ , S µi ] − T µ − i S µi . We note that[Λ , S µi ] − T µ − i S µi = [Λ ,S µi − S i | ] − ( T µ − i − T i | )( S µi − S i | ) + [Λ , S i | ] − T i | ( S µi − S i | ) − ( T µ − i − T i | ) S i | − T i | S i | = [Λ , S µi − S i | ] − ( T µ − i − T i | )( S µi − S i | ) − T i | ( S µi − S i | ) − ( T µ − i − T i | ) S i | (2.3.12)We note that since [Λ , S µi ] − T µ − i S µi ≡ µ −
0, ( T µ − i − T i | ) S i | contributes nothing to [Λ , S µi ] − S µi T µ − i . Let us estimate [Λ , S µi − S i | ] in (2 . . = P n +1 α,β =1 Λ iαβ ( x i ) ∂∂x αi ∧ ∂∂x βi with Λ iαβ ( x i ) = − Λ iβα ( x i ), where x i = ( w i , z i ). We may assume that | Λ iαβ ( x i ) | < M for some positive constant M > , S µi − S i | ] = P n +1 α,β =1 iαβ ( x i ) ∂ ( S µi − S i | ) ∂x αi ∂∂x βi . Let B i ( z i ) := S µi − S i | . Then ∂B i ( z i ) ∂w i = 0 and ∂B i ∂z αi = πi R | ξ − z αi | = δ B i ( z i ,..., α − th ξ ,...,z ni )( ξ − z αi ) dξ ≪ A ( t ) δ for ( z i , w i ) for | z i | < − δ, | w i | <
1. Then we obtain[Λ , S µi − S i | ] = n +1 X α,β =1 iαβ ( x i ) ∂B i ( z i ) ∂x αi ∂∂x βi ≪ n + 1) M A ( t ) δ (2.3.13)Hence we have, from (2 . . . . . . . .
10) and (2 . . , S µi ] − T µ − i S µi ≪ n + 1) Mδ A ( t ) + d A ( t ) + d A ( t ) ≪ d A ( t ) , p ∈ U δi , (2.3.14)where d := 2( n + 1) Mδ + d bc + d = d + d bc , where d := 2( n + 1) Mδ + d . (2.3.15)We claim that T i | µ ( t ) ≪ d ǫ A ( t ) , p ∈ U δi (2.3.16)Indeed, from S i | ( p ) = w i ( p ), (2 . .
11) and (2 . . | w i ( p ) | ≥ ǫ , T i | µ ( t ) ≪ d ǫ A ( t ). If | w i ( p ) | < ǫ , we get T i | µ ( t ) ≪ d ǫ A ( t ) by the maximum principle.On the other hand, from (2 . . d ǫ = d ǫ + d bǫc . Now we set d = d ǫ . If we take c > bǫ , (2.3.17)then we get d ǫ < d ǫ + d ǫ = d so that we obtain (2 . .
9) from (2 . .
7) and (2 . . EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 15
Lastly we show that S µ +1 i ( t ) − S i | ≪ A ( t ) , p ∈ U i (2.3.18)We note that (for the detail, see [KS59] p.493) ψ ik | µ +1 ( t ) ≪ bc c A ( t ) , p ∈ V ∩ U i ∩ U k (2.3.19)Recall from (2 . .
25) that W i | µ +1 ( p, t ) ≡ µ +1 [Λ , S µi ( p, t )] | V − S µi ( p, t ) T µi ( p, t ). Since [Λ , S µi ] − S µi T µi ≡ µ , S µi ] − S µi T µi ≡ µ +1 [Λ , S µi ] − ( S µi − S i | )( T µi − T i | ) − S µi T i | − S i | ( T µi − T i | ). Since [Λ , S µi ] − S µi T µi ≡ µ
0, we obtain, from (2 . .
6) and (2 . . W i | µ +1 ( p, t ) ≪ bd c A ( t ) , p ∈ V ∩ U δi . (2.3.20) Lemma 2.3.21 (compare [KS59] p.499) . We can choose φ i | µ +1 ( p, t ) satisfying ψ ik | µ +1 ( p, t ) = φ i | µ +1 ( p, t ) − f ik | ( p ) φ k | µ +1 ( p, t ) W i | µ +1 ( p, t ) = − [Λ , φ iµ +1 ( p, t )] | V + φ i | µ +1 T i | ( p ) such that φ i | µ +1 ≪ c (cid:0) bc c + bd c (cid:1) A ( t ) , where the constant c is independent of µ .Proof. For simplicity, we write U i for V ∩ U i , U δi for V ∩ U δi , and let U = { U i } be the covering of V . Forany 0-cochain φ = { φ i ( p ) } , 1-cochain ψ = { ψ ik ( p ) , W i ( p ) } on U , we define the norms of φ, ( ψ, W ) by || φ || := max i sup p ∈ U i | φ i ( p ) | , || ( ψ, W ) || := max i,k sup p ∈ U i ∩ U k | ψ ik ( p ) | + max i sup p ∈ U δi | W i ( p ) | The coboundary φ is defined by f ik ( p ) φ k ( p ) − φ i ( p ) , p ∈ U i ∩ U k , − [ φ i ( p ) , Λ ] | V + φ i ( p ) T i | ( p ) , p ∈ U i For any ( ψ, W ), we define ι ( ψ, W ) = inf δ ( φ )=( ψ,W ) || φ || It suffices to prove the existence of constant c such that ι ( ψ, W ) ≤ c || ( ψ, W ) || . Assume that such a constant c does not exist. Then we can find a sequence ( ψ ′ , W ′ ) , ( ψ ′′ , W ′′ ) , · · · , ( ψ µ , W µ ) , · · · such that ι ( ψ ( µ ) , W ( µ ) ) = 1 , || ( ψ ( µ ) , W ( µ ) ) || < µ We take a covering { ¯ U δi } of V . Since φ µk ( p ) < p ∈ U k , there exists a subsequence φ ( µ ) , φ ( µ ) , · · · , φ ( µ v ) of φ ′ , φ ′′ , · · · such that φ ( µ v ) k converges absolutely and uniformly on ¯ U δk for each k . Since V is compact, wecan choose a subsequence that works for all k . On the other hand, since || ( ψ, W ) || < µ , we have in particular | f ik | ( p ) φ ( µ ) k ( p ) − φ ( µ ) i ( p ) | < µ , p ∈ U i ∩ U k , | − [ φ ( µ ) i ( p ) , Λ ] | V + φ ( µ ) i ( p ) T i | ( p ) | < µ , p ∈ U δi (2.3.22)Then φ ( µ v ) i converges absolutely and uniformly on the whole U i . Let φ i ( p ) = lim v φ ( µ v ) i ( p ) and let φ = { φ i ( p ) } . Then we have || φ ( µ v ) − φ || → n → ∞ . On the other hand, from (2 . . δφ = (0 , W φ ), where W φ ( p ) = 0 for p ∈ U δi . By identity theorem, W φ ( p ) = 0 for p ∈ U i . Hence we have δ ( φ ( µ v ) − φ ) = ( ψ ( µ v ) , W ( µ v ) ) which contradicts to ι ( ψ ( µ v ) , W ( µ v ) ) = 1. (cid:3) By Lemma 2.3.21, we can choose S i | µ +1 ( t ) ≪ c (cid:0) bc c + bd c (cid:1) A ( t ). We note (2 . .
8) and (2 . . c > max { bc (1 + c ) , bǫ , c bc + c bd } , we get (2 . . b, c, c , d are independent of µ ,we have (2 . . , (2 . . . . N = { t | P mr =1 | t r | < c m } , the power series S i ( p, t ) , f ik ( p, t )and T i ( p, t ) converges absolutely and uniformly for t ∈ N so that S i ( p, t ) , T i ( p, t ) and f ik ( p, t ) are holomorphicon U δi × N and U δi ∩ U δk × N , respectively, and satisfy (2 . . , (2 . . , (2 . . , (2 . .
7) and (2 . .
8) by replacing U i by U δi . This completes the proof of Theorem 2.2.3. (cid:3) Maximal families: Theorem of completeness.Definition 2.4.1.
Let
V ⊂ ( W × M, Λ ) ω −→ M be a Poisson analytic family of compact holomorphic Poissonsubmanifolds of ( W, Λ) of codimension and let t be a point on M . We say that V ω −→ M is maximal at t if, for any Poisson analytic family V ′ ⊂ ( W × M ′ , Λ) ω ′ −→ M ′ of compact holomorphic Poisson submanifoldsof ( W, Λ ) of codimension such that ω − ( t ) = ω ′− ( t ′ ) , t ′ ∈ M ′ , there exists a holomorphic map h of aneighborhood N ′ of t ′ on M ′ into M which maps t ′ to t such that ω ′− ( t ′ ) = ω − ( h ( t ′ )) for t ′ ∈ N ′ . Wenote that if we set a Poisson map ˆ h : ( W × N ′ , Λ ) → ( W × M, Λ ) defined by ( w, t ′ ) → ( w, h ( t ′ )) , then therestriction map of ˆ h to V ′ | N ′ = ω ′− ( N ′ ) ⊂ ( W × N ′ , Λ) defines a Poisson map V ′ | N ′ → V so that V ′ | N ′ isthe family induced from V by h , which means V ω −→ M is complete at t . Theorem 2.4.2 (Theorem of completeness) . Let
V ⊂ ( W × M, Λ ) ω −→ M be a Poisson analytic family ofcompact holomorphic Poisson submanifolds of ( W, Λ ) of codimension . If the characteristic map ρ d, : T M → H ( V , N • | V ) is an isomorphism, then the family V ω −→ M is maximal at the point t = 0 .Proof. We extend the arguments in [KS59] p.494-496 in the context of holomorphic Poisson deformations.We tried to maintain notational consistency with [KS59].Suppose that M = { t | P mr =1 | t r | < } and that ρ d, : T M → H ( V , N • | V ) is an isomorphism. Let V ′ ⊂ ( W × M, Λ ) ω ′ −→ M ′ be an arbitrary Poisson analytic family of holomorphic Poisson submanifoldsof ( W, Λ ) of codimension 1 such that ω ′− (0) = V , where M ′ = { s | P lr =1 | s r | < } . We will constructa holomorphic map h : s → t = h ( s ) of N ′ into M with h (0) = 0 such that ω ′− ( s ) = ω − ( h ( s )) where N ′ = { s | P lr =1 | s r | < δ } ⊂ M ′ for a sufficiently small number δ > { S i ( p, t ) } , { f ik ( p, t ) } , and { T i ( p, t ) } determine the Poissonanalytics family V . Let { R i ( p, s ) } , { e ik ( p, s ) } and { Q i ( p, t ) } and be the corresponding system defining V ′ ⊂ ( W × M ′ , Λ ) so that we have S i ( p, t ) = f ik ( p, t ) S k ( p, t ) , [Λ , S i ( p, t )] = S i ( p, t ) T i ( p, t )(2.4.3) R i ( p, s ) = e ik ( p, s ) R k ( p, s ) , [Λ , R i ( p, s )] = R i ( p, s ) Q i ( p, s )(2.4.4)We may assume that S i ( p,
0) = R i ( p,
0) = w i ( p ) , f ik ( p,
0) = e ik ( p,
0) = f ik | ( p ) , T i ( p,
0) = R i ( p,
0) = T i | ( p ) . (2.4.5)We expand S i ( p, t ) = w i ( p )+ S i | ( p, t )+ S i | ( p, t )+ · · · and let S i | ( p, t ) = P mr =1 B ir ( p ) t r . Then the restriction β ir ( p ) of B ir ( p ) to V satisfy β ir ( p ) = f ik | ( p ) β kr ( p ) , p ∈ V ∩ U i ∩ U k , (2.4.6) − [ β ir ( p ) , Λ ] | V + β ir ( p ) T i | ( p ) = 0 , p ∈ V ∩ U i (2.4.7)and { β , ..., β m } forms a basis of H ( V , N • | V ) by the hypothesis.If there exist non-vanishing holomorphic functions f i ( p, s ) defined on U i × N ′ satisfying f i ( p, s ) R i ( p, s ) = S i ( p, h ( s )) , (2.4.8)we get ω ′− ( s ) = ω − ( h ( s )). Recall Notation 1 and let us write h ( s ) and f i ( p, s ) in the following form: h ( s ) = ( h ( s ) = ∞ X µ =1 h r | µ ( s ) , ..., h m ( s ) = ∞ X µ =1 h r | µ ( s )) , f i ( p, s ) = 1 + ∞ X µ =1 f i | µ ( p, s ) , We will construct such f i ( p, s ) and h ( s ) satisfying (2 . .
8) by solving the system of congruences by inductionon µ f µi ( p, s ) R i ( p, s ) ≡ µ S i ( p, h µ ( s )) , µ = 0 , , , · · · . (2.4.9)(2 . . follows from (2 . . h µ − ( s ) and f µ − i ( p, s ) satisfying (2 . . µ − are alreadydetermined. We will find h µ ( s ) and f i | µ ( p, s ) such that h µ := h µ − ( s ) + h µ ( s ) and f µi ( p, s ) := f µ − i ( p, s ) + f i | µ ( p, s ) satisfy (2 . . µ . We can define homogenous polynomials Γ i | µ ( p, s ) of degree µ in s byΓ i | µ ( p, s ) ≡ µ f µ − i ( p, s ) R i ( p, s ) − S i ( p, h µ − ( s )) , p ∈ U i (2.4.10) EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 17
Then we claim that Γ i | µ ( p, s ) = f ik | ( p )Γ k | µ ( p, s ) , p ∈ V ∩ U i ∩ U k (2.4.11) [Λ , Γ i | µ ( p, s )] | w i ( p )=0 = Γ i | µ ( p, s ) T i | ( p ) , p ∈ V ∩ U i (2.4.12)Indeed, (2.4.11) follows from [KS59] p.496. On the other hand, to prove (2 . . , f µ − i ( p, s )] + Q i ( p, s ) f µ − i ( p, s ) − f µ − i ( p, s ) T i ( p, h µ − ( p, s )) ≡ µ − , − ] on f µ − i ( p, s ) R i ( p, s ) ≡ µ − S i ( p, h µ − ( s )) in (2 . . µ − , we get, from (2 . . . . . . µ − ,[Λ , f µ − i ( p, s )] R i ( p, s ) + [Λ , R i ( p, s )] f µ − i ( p, s ) ≡ µ − [Λ , S i ( p, h µ − ( s ))] ⇐⇒ [Λ , f µ − i ( p, s )] R i ( p, s ) + R i ( p, s ) Q i ( p, s ) f µ − i ( p, s ) ≡ µ − S i ( p, h µ − ( s )) T i ( p, h µ − ( s )) ≡ µ − f µ − i ( p, s ) R i ( p, s ) T i ( p, h µ − ( s )) ⇐⇒ ([Λ , f µ − i ( p, s )] + Q i ( p, s ) f µ − i ( p, s ) − f µ − i ( p, s ) T i ( p, h µ − ( p, s ))) R i ( p, s ) ≡ µ − . . . . . . µ , (2 . . . . , Γ i | µ ( p, s )] ≡ µ [Λ , f µ − i ( p, s ) R i ( p, s )] − [Λ , S i ( p, h µ − ( s ))](2.4.14) ≡ µ [Λ , f µ − i ( p, s )] R i ( p, s ) + [Λ , R i ( p, s )] f µ − i ( p, s ) − [Λ , S i ( p, h µ − ( s ))] ≡ µ [Λ , f µ − i ( p, s )] R i ( p, s ) + R i ( p, s ) Q i ( p, s ) f µ − i ( p, s ) − S i ( p, h µ − ( s )) T i ( p, h µ − ( s )) ≡ µ [Λ , f µ − i ( p, s )] R i ( p, s ) + R i ( p, s ) Q i ( p, s ) f µ − i ( p, s ) + Γ i | µ ( p, s ) T i | ( p ) − f µ − i ( p, s ) R i ( p, s ) T i ( p, h µ − ( s )) ≡ µ ([Λ , f µ − i ( p, s )] + Q i ( p, s ) f µ − i ( p, s ) − f µ − i ( p, s ) T i ( p, h µ − ( p, s ))) R i ( p,
0) + Γ i | µ ( p, s ) T i | ( p )By restricting (2 . .
14) to V (i.e by setting S i ( p,
0) = R i ( p,
0) = w i ( p )), we obtain[Λ , Γ i | µ ( p, s )] | w i ( p )=0 ≡ µ Γ i | µ ( p, s ) T i | ( p ) , p ∈ V ∩ U i . (2.4.15)This proves (2 . . . .
11) and (2 . . h r | µ of degree µ in s such that m X r =1 β ir ( p ) h r | µ ( s ) = Γ i | µ ( p, s ) , p ∈ V ∩ U i (2.4.16)From h µr ( s ) = h µ − r ( s ) + h r | µ ( s ), the congruence (2 . . µ is equivalent to (for the detail, see [KS59] p.496) m X r =1 B ir ( p ) h r | µ ( s ) = w i ( p ) f i | µ ( p, s ) + Γ i | µ ( p, s )(2.4.17)By setting f i | µ ( p, s ) := P mr =1 B ir ( p ) h r | µ ( s ) − Γ i | µ ( p,s ) w i ( p ) , we get (2 . . h µ ( s ) and f µi ( p, s ) satisfying (2 . . µ . (cid:3) Proof of convergence.
The convergence of h r ( s ) , f i ( p, s ) follows from the same arguments in [KS59] p.497-498, which completesTheorem 2.4.2.3. Deformations of compact holomorphic Poisson submanifolds of arbitrary codimensions
We extend Definition 2 . .
26 to arbitrary codimensions.
Definition 3.0.1 (compare [Kod62]) . Let ( W, Λ ) be a holomorphic Poisson manifold of dimension d + r .We denote a point in W by w and a local coordinate of w by ( w , ..., w r + d ) . By a Poisson analytic familyof compact holomorphic Poisson submanifolds of ( W, Λ ) we mean a holomorphic Poisson submanifold V ⊂ ( W × M, Λ ) of codimension r , where M is a complex manifold and Λ is the holomorphic Poisson structureon W × M induced from ( W, Λ ) , such that (1) for each point t ∈ M , V t × t := ω − ( t ) = V ∩ π − ( t ) is a connected compact holomorphic Poissonsubmanifold of ( W × t, Λ ) of dimension d , where ω : V → M is the map induced from the canonicalprojection π : W × M → M . (2) for each point p ∈ V , there exist r holomorphic functions f α ( w, t ) , α = 1 , ..., r defined on a neighbor-hood U p of p in W × M such that rank ∂ ( f ,...,f r ) ∂ ( w ,...w r + d ) = r , and U p ∩ V is defined by the simultaneousequations f α ( w, t ) = 0 , α = 1 , ..., r .We call V ⊂ ( W × M, Λ ) a Poisson analytic family of compact holomorphic Poisson submanifolds V t , t ∈ M of ( W, Λ ) . We also call V ⊂ ( W × M, Λ ) a Poisson analytic family of deformations of a compact holomorphicPoisson submanifold V t of ( W, Λ ) for each fixed point t ∈ M . The complex associated with the normal bundle of a holomorphic Poisson submanifold ina holomorphic Poisson manifold.
Let ( W, Λ ) be a holomorphic Poisson manifold and V be a holomorphic Poisson submanifold of ( W, Λ ).Let U = { W i } be a open covering such that W i is a polycylinder with a local coordinate ( w i , z i ) =( w i , ..., w ri , z i , ..., z di ) such that W i = { ( w i , z i ) || w i | < , | z i | < } , where | w i | = max λ | w λi | , | z i | = max α | z αi | ,the local coordinate ( w i , z i ) can be extended to a domain containing the closure of W i and on each neighbor-hood W i , V ∩ W i coincides with the subspace of W i determined by w i = · · · = w ri = 0. On the intersection W i ∩ W k , the coordinates w i , ..., w ri , z i , ..., z di are holomorphic functions of w k and z k : w αi = f αik ( w k , z k ) , α =1 , ..., r, z λi = g λik ( w k , z k ) , λ = 1 , ..., d . We set f ik ( w k , z k ) := ( f ik ( w k , z k ) , · · · f rik ( w k , z k )) and g ik ( w k , z k ) =( g ik ( w k , z k ) , · · · , g dik ( w k , z k )) so that we write the formula in the form w i = f ik ( w k , z k ) , z i = g ik ( w k , z k ).Then we have f ik (0 , z k ) = 0 and so w αi = f αik ( w k , z k ) has the following form: w αi = f αik ( w k , z k ) = r X β =1 w βk F αikβ ( w k , z k )(3.1.1)We set U i = V ∩ W i = { (0 , z i ) || z i | < } . We denote a point of V by z and if z = (0 , z i ) ∈ U i , we consider z i = ( z i , ..., z di ) as the coordinate of z on U i . We indicate by writing z = (0 , z k ) ∈ U k ∩ U i that z is a pointin U k ∩ U i whose coordinate on U k is z k . We note that F αikβ (0 , z k ) = ∂f αik ( w k , z k ) ∂w βk | w k =0 , z = (0 , z k ) ∈ U i ∩ U k , (3.1.2)and let F ik ( z ) := ( F αikβ (0 , z )) α,β =1 ,...,r . Then the matrix valued functions F ik ( z ) satisfy F ik ( z ) = F ij ( z ) F jk ( z )for z ∈ U i ∩ U j ∩ U k so that they define the normal bundle N V/W .On the other hand, since V is a holomorphic Poisson submanifold of ( W, Λ ), [Λ , w αi ] is of the form[Λ , w αi ] = r X β =1 w βi T βiα ( w i , z i )(3.1.3)where T βiα ( w i , z i ) = P rγ =1 P iαβ ( w i , z i ) ∂∂w i + · · · + P riαβ ( w i , z i ) ∂∂w ri + Q iαβ ( w i , z i ) ∂∂z i + · · · + Q diαβ ( w i , z i ) ∂∂z di ∈ Γ( W i , T W ) by which we consider T βiα ( w i , z i ) a vector-valued holomorphic function on W i . Then on W i ∩ W k ,we have [Λ , w αi ] = r X β =1 w βi T βiα ( w i , z i ) = r X β =1 f βik ( w k , z k ) T βiα ( w i , z i )(3.1.4)On the other hand, from (3 . .
1) and (3 . . , w αi ] = r X β =1 w βk [Λ , F αikβ ( w k , z k )] + r X β =1 F αikβ ( w k , z k )[Λ , w βk ](3.1.5) = r X β =1 w βk [Λ , F αikβ ( w k , z k )] + r X β,γ =1 F αikβ ( w k , z k ) w γk T γkβ ( w k , z k )By taking the derivative of (3 . .
4) and (3 . .
5) with respect to w γk and setting w k = 0, we get from (3 . . U i ∩ U k , T W | V ), r X β =1 F βikγ (0 , z k ) T βiα (0 , z i ) = [Λ , F αikγ (0 , z k )] | w k =0 + r X β =1 F αikβ (0 , z k ) T γkβ (0 , z k )(3.1.6) EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 19
On the other hand, from (3 . . , [Λ , w αi ]] = r X β =1 [Λ , w βi T βiα ( w i , z i )] = r X β =1 w βi [Λ , T βiα ( w i , z i )] − [Λ , w βi ] ∧ T βiα ( w i , z i )= r X β =1 w βi [Λ , T βiα ( w i , z i )] − r X β,γ =1 w γi T γiβ ( w i , z i ) ∧ T βiα ( w i , z i )= r X β =1 w βi [Λ , T βiα ( w i , z i )] − r X γ =1 T βiγ ( w i , z i ) ∧ T γiα ( w i , z i ) ! (3.1.7)By taking the derivative of (3 . .
7) with respect to w βi and setting w i = 0, we get, on Γ( U i , T W | V ),[Λ , T βiα (0 , z i )] | w i =0 − r X γ =1 T βiγ (0 , z i ) ∧ T γiα (0 , z i ) = 0(3.1.8)Now we define a complex of sheaves associated with the normal bundle N V/W : N V/W ∇ −→ N V/W ⊗ T W | V ∇ −→ N V/W ⊗ ∧ T W | V ∇ −→ N V/W ⊗ ∧ T W | V ∇ −→ · · · First we define ∇ : N V/W → N
V/W ⊗ T V | W and then extend to ∇ : N V/W ⊗∧ p T W | V → N V/W ⊗∧ p +1 T W | V in the following. We note that Γ( U i , N W/V ) ∼ = ⊕ r Γ( U i , O V ) and Γ( U i , N V/W ⊗ T W | V ) ∼ = ⊕ r Γ( U i , T W | V ).Using these isomorphism, we define ∇ on N V/W by the rule ∇ ( e αi ) := r X β =1 T αiβ (0 , z i ) e βi where e αi = (0 , · · · , α − th , · · · , ∈ ⊕ ri =1 Γ( U i , O V ). In general, we define ∇ : ⊕ r Γ( U i , O V ) → ⊕ r Γ( U i , T W | V ) r X α =1 g αi e αi r X α =1 − [ g αi , Λ ] | w i =0 · e αi + r X α =1 g αi ∇ ( e αi ) = r X α =1 − [ g αi , Λ ] | w i =0 + r X β =1 g βi T βiα (0 , z i ) e αi where g αi ∈ Γ( U i , O V ).We extend ∇ on N V/W ⊗ ∧ p T W | V . N V/W ⊗ ∧ p T W | V ∇ −→ N V/W ⊗ ∧ p +1 T W | V is locally defined in thefollowing way: we note that Γ( U i , N V/W ⊗ ∧ p T W | V ) ∼ = ⊕ r Γ( U i , ∧ p T W | V ) and Γ( U i , N V/W ⊗ ∧ p +1 T W | V ) ∼ = ⊕ r Γ( U i , ∧ p +1 T W | V ). From these isomorphism, we define ∇ by the rule ∇ : ⊕ r Γ( U i , ∧ p T W | V ) → ⊕ r Γ( U i , ∧ p +1 T W | V ) r X α =1 g αi e αi r X α =1 − [ g αi , Λ ] | w i =0 · e αi + ( − p r X α =1 g αi ∧ ∇ ( e αi )= r X α =1 − [ g αi , Λ ] | w i =0 + ( − p r X β =1 g βi ∧ T βiα (0 , z i ) e αi where g αi ∈ Γ( U i , ∧ p T W | V ). First we show that ∇ defines a complex, i.e. ∇ ◦ ∇ = 0. Indeed, ∇ ◦ ∇ ( r X α =1 g αi e αi ) = ∇ ( r X α =1 − [ g αi , Λ ] | w i =0 + ( − p r X β =1 g βi ∧ T βiα (0 , z i ) e αi )= − [( − p r X β =1 g βi ∧ T βiα (0 , z i ) , Λ ] | w i =0 · e αi + ( − p +1 r X α =1 − [ g αi , Λ ] | w i =0 + ( − p r X β =1 g βi ∧ T βiα (0 , z i ) ∇ ( e αi )= ( − p +1 r X α,β =1 [ g βi ∧ T βiα (0 , z i ) , Λ ] | w i =0 e αi + ( − p r X α,β =1 [ g βi , Λ ] | w i =0 ∧ T βiα (0 , z i ) e αi − r X α,β,γ =1 g βi ∧ T βiγ (0 , z i ) ∧ T γα (0 , z i ) e αi Hence in order to show ∇ ◦ ∇ = 0, we have to show that( − p +1 r X β =1 [ g βi ∧ T βiα (0 , z i ) , Λ ] | w i =0 + ( − p r X β =1 [ g βi , Λ ] | w i =0 ∧ T βiα (0 , z i ) − r X β,γ =1 g βi ∧ T βiγ (0 , z i ) ∧ T γiα (0 , z i ) = 0(3.1.9)Indeed, from (3.1.8), (3.1.9) becomes( − p +1 r X β =1 [ g βi , Λ ] | w i =0 ∧ T βiα (0 , z i ) + ( − p +1+ p r X β =1 g βi ∧ [ T βiα (0 , z i ) , Λ ] | w i =0 +( − p r X β =1 [ g βi , Λ ] | w i =0 ∧ T βiα (0 , z i ) − r X β,γ =1 g βi ∧ T βiγ (0 , z i ) ∧ T γiα (0 , z i )= r X β =1 g βi ∧ [Λ , T βiα (0 , z i )] | w i =0 − r X β,γ =1 g βi ∧ T βiγ (0 , z i ) ∧ T γiα (0 , z i )= r X β =1 g βi [Λ , T βiα (0 , z i )] | w i =0 − r X γ =1 T βiγ (0 , z i ) ∧ T γiα (0 , z i ) ! = 0Hence ∇ ◦ ∇ = 0.Next we show ∇ is well-defined. In other words, on U i ∩ U k , the following diagram commutes(3.1.10)Γ( U k , N W/V ⊗ ∧ p T W/V ) ∼ = ⊕ r Γ( U k , ∧ p T W | V ) on U i ∩ U k −−−−−−−→ ∼ = Γ( U i , N W/V ⊗ ∧ p T W/V ) ∼ = ⊕ r Γ( U i , ∧ p T W | V ) ∇ y y ∇ Γ( U k , N W/V ⊗ ∧ p +1 T W/V ) ∼ = ⊕ r Γ( U k , ∧ p +1 T W/V ) on U i ∩ U k −−−−−−−→ ∼ = Γ( U i , N W/V ⊗ ∧ p +1 T W/V ) ∼ = ⊕ r Γ( U i , ∧ p +1 T W/V )Let P rα =1 g αk e αk ∈ ⊕ r Γ( U k , ∧ p T W | V ). Then ∇ ( P rα =1 g αk e αk ) = P rα =1 ( − [ g αk , Λ ] | w k =0 + ( − p P rβ =1 g βk ∧ T βkα (0 , z k )) e αk is identified on U i ∩ U k with r X γ =1 r X α =1 − F γikα (0 , z k )[ g αk , Λ ] | w k =0 + ( − p r X α,β =1 F γikα (0 , z k ) g βk ∧ T βkα (0 , z k ) e γi (3.1.11)On the other hand, P rα =1 g αk e αk is identified on U i ∩ U k with P rα,γ =1 ( F γikα (0 , z k ) g αk ) e γi ∈ ⊕ r Γ( U i , ∧ p T W | V )and ∇ ( r X α,γ =1 ( F γikα (0 , z k ) g αk ) e γi ) = r X γ =1 r X α =1 − [ F γikα (0 , z k ) g αk , Λ ] | w k =0 + ( − p r X α,β =1 F βikα (0 , z k ) g αk ∧ T βiγ (0 , z i ) e γi (3.1.12) EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 21
Hence in order for the diagram (3.1.10) to commute, we have to show that (3.1.11) coincides with (3.1.12):( − p r X α,β =1 F γikα (0 , z k ) g βk ∧ T βkα (0 , z k ) = r X α =1 − [ F γikα (0 , z k ) , Λ ] | w k =0 ∧ g αk + ( − p r X α,β =1 F βikα (0 , z k ) g αk ∧ T βiγ (0 , z i ) = 0 ⇐⇒ r X α,β =1 F γikβ (0 , z k ) T αkβ (0 , z k ) ∧ g αk = r X α =1 − [ F γikα (0 , z k ) , Λ ] | w k =0 ∧ g αk + r X α,β =1 F βikα (0 , z k ) T βiγ (0 , z i ) ∧ g αk = 0which follows from (3 . . ∇ is well-defined. Definition 3.1.13.
We call the complex defined as above N • V/W : N V/W ∇ −→ N V/W ⊗ T W | V ∇ −→ N V/W ⊗ ∧ T W | V ∇ −→ N V/W ⊗ ∧ T W | V ∇ −→ · · · the complex associated with the normal bundle N V/W of a holomorphic Poisson submanifold V of a holo-morphic Poisson manifold W and denote its i -th hypercohomology group by H i ( V, N • V/W ) . Example 1. On W = C , let ( z, w , w ) be a coordinate and Λ = w z ∂∂w ∧ ∂∂w be a holomorphic Poissonstructure on C . Then w = w = 0 is a holomorphic Poisson submanifold V = C with coordinate z so thatthe normal bundle is N V/W ∼ = O C ⊕ O C . Then [Λ , w ] = w z ∂∂w and [Λ , w ] = w ( − z ∂∂w ) so that we get T ( z ) = z ∂∂w , T ( z ) = 0 and T ( z ) = − z ∂∂w , T ( z ) = 0 . Since [Λ , f i ( z )] | w = w =0 = 0 for entire functions f i ( z ) , i = 1 , , we have ∇ ( f e + f e ) = ( f T ( z )+ f T ( z )) e +( f T ( z )+ f T ( z )) e = ( f z ∂∂w , − f z ∂∂w ) so that H ( V, N • V/W ) = { (0 , f ( z )) | f ( z ) is an entire function } . Infinitesimal deformations.
Let M = { t = ( t , ..., t l ) ∈ C l || t | < } . Consider a Poisson analytic family V ⊂ ( W × M , Λ ) ofcompact holomorphic Poisson submanifolds V t , t ∈ M of ( W, Λ ) and let V = V as in Definition 3 . .
1. Wekeep the notations in subsection 3.1. Let ǫ be a sufficiently small positive number. Then for | t | < ǫ , theholomorphic Poisson submanifold V t of W is defined in each neighborhood W i by simultaneous equations ofthe form w λi = ϕ λi ( z i , t ) , λ = 1 , ..., r where the ϕ λi ( z i , t ) are holomorphic functions of z i , | z i | <
1, dependingholomorphically on t , | t | < ǫ , and satisfying the boundary conditions ϕ λi ( z i ,
0) = 0 , λ = 1 , ..., r.
By setting ϕ i ( z i , t ) = ( ϕ i ( z i , t ) , · · · , ϕ ri ( z i , t )), we write the simultaneous equation as w i = ϕ i ( z i , t ). Then we have ϕ i ( g ik ( ϕ k ( z k , t ) , z k ) , t ) = f ik ( ϕ k ( z k , t ) , z k ). For each t , | t | < ǫ , we set w λti = w λi − ϕ λi ( z i , t ) , λ = 1 , .., r sothat ( w ti , ..., w rti , z i , ..., z di ) form a local coordinate defined on W i . We define F tik ( z t ) := (cid:16) ∂w λti ∂w µtk ( z t ) (cid:17) λ,µ =1 ,...,r for z t ∈ V t ∩ W i ∩ W k , where we denote by ( ∂w λti ∂w µtk ( z t )) the value of the partial derivative ∂w λti ∂w µtk at a point z t on V t . Then the collection { F tik ( z t ) } of F tik ( z t ) forms a system of transition matrices defining the normalbundle F t of V t in W . Note that F ik ( z ) = F ik ( z ) for z ∈ U i ∩ U k from (3 . . ∂∂t = P ρ γ ρ ∂∂t ρ of M at t , | t | < ǫ , and let ψ i ( z t , t ) = ∂ϕ i ( z i ,t ) ∂t for z t = ( ϕ i ( z i , t ) , z i ). Then we obtain the equality ψ i ( z t , t ) = F tik ( z t ) · ψ k ( z t , t ) , for z t ∈ V t ∩ W i ∩ W k . (3.2.1)On the other hand, w i − ϕ i ( z i , t ) = 0 define a holomorphic Poisson submanifold, we have[Λ , w λi − ϕ λi ( z i , t )] = r X µ =1 ( w µi − ϕ µi ( z i , t )) T µiλ ( w, z i , t )(3.2.2)for some T µi ( w i , z i , t ) which is of the form T µiλ ( w i , z i , t ) = P µi ( w i , z i , t ) ∂∂w i + · · · + P µir ( w i , z i , t ) ∂∂w ri + Q µi ( w i , z i , t ) ∂∂z i + · · · + Q µid ( w i , z i , t ) ∂∂z di by which we consider T µiλ ( w i , z i , t ) as a vector valued holomorphic function of ( w i , z i , t ).By taking the derivative of (3.2.2) with respect to t , we get [Λ , − ∂ϕ λi ( z i ,t ) ∂t ] = P rµ =1 − ∂ϕ µi ( z i ,t ) ∂t T µiλ ( w, z i , t )+ P rµ =1 ( w λi − ϕ µi ( z i , t )) ∂T µiλ ( w,z i ,t ) ∂t . By restricting to V t , equivalently setting w i − ϕ i ( z i , t ) = 0, we get, on Γ( W i ∩ V t , T W | V t ), − [Λ , ∂ϕ λi ( z i , t ) ∂t ] | V t + r X µ =1 ∂ϕ µi ( z i , t ) ∂t T µiλ ( ϕ i ( z i , t ) , z i , t ) = 0 ⇐⇒ − [ ψ λi ( z i , t ) , Λ ] | V t + n X µ =1 ψ µi ( z i , t ) T µiλ ( ϕ i ( z i , t ) , z i , t ) = 0(3.2.3)From (3.2.1) and (3.2.2), { ψ i ( z i , t ) } defines an element in H ( V t , N • V/W t ) so that we have a linear map σ t : T t ( M ) → H ( V t , N • V t /W ) ∂∂t ∂V t ∂t := { ψ i ( z t , t ) } We call σ t the characteristic map. Example 2.
Let [ ξ , ξ , ξ , ξ ] be the homogenous coordinate on P C . Let [1 , z , z , z ] = [1 , ξ ξ , ξ ξ , ξ ξ ] . Then V ⊂ ( P C × C , Λ = z ∂∂z ∧ ∂∂z ) defined by ξ = ξ − tξ = 0 is a Poisson analytic family of deformations of aholomorphic Poisson submanifold P C ∼ = V : ξ = ξ = 0 of ( P , Λ ) . We note that N P C / P C ∼ = O P C (1) ⊕ O P C (1) so that we have the characteristic map T C → H ( V, N • V/ P C ) = H ( P C , ( O P C (1) ⊕ O P C (1)) • ) a (0 , a ) = (0 + 0 · z , a + 0 · z )3.3. Theorem of existence.Theorem 3.3.1 (theorem of existence) . Let V be a compact holomorphic Poisson submanifold of a holo-morphic Poisson manifold ( W, Λ ) . If H ( V, N • V/W ) = 0 , then there exists a Poisson analytic family V ofcompact holomorphic Poisson submanifolds V t , t ∈ M of ( W, Λ ) such that V = V and the characteristicmap σ : T ( M ) → H ( V, N • V/W ) ∂∂t (cid:18) ∂V t ∂t (cid:19) t =0 is an isomorphism.Proof. We extend the arguments in [Kod62] p.150-158 in the context of holomorphic Poisson deformations.We tried to keep notational consistency with [Kod62]. We also keep the notations in subsection 3.1.Let { γ , ..., γ ρ , ..., γ l } be a basis of H ( V, N • V/W ), where l = dim H ( V, N • V/W ). On each neighborhood U i = V ∩ W i , γ ρ is represented as a vector-valued holomorphic function γ ρi = ( γ ρi ( z i ) , ..., γ αρi ( z i ) , ..., γ rρi ( z i )) ∈ ⊕ r Γ( U i , O V )such that γ ρi ( z ) = F ik ( z ) · γ ρk ( z ) , z ∈ U i ∩ U k (3.3.2) − [Λ ,γ αρi ( z i )] | w i =0 + r X β =1 γ βρi ( z i ) T βiα (0 , z i ) = 0 , z i ∈ U i , α = 1 , ..., r. (3.3.3)Let ǫ be a small positive number. In order to prove Theorem 3.3.1, it suffices to construct vector-valuedholomorphic functions ϕ i ( z i , t ) = ( ϕ i ( z i , t ) , ..., ϕ ri ( z i , t ))in z i , and t with | z i | < , | t | < ǫ , with | ϕ i ( z i , t ) | < ϕ i ( z i ,
0) = 0 , ∂ϕ i ( z i , t ) ∂t ρ | t =0 = γ ρi ( z )such that ϕ i ( g ik ( ϕ k ( z k , t ) , z k ) , t ) = f ik ( ϕ ( z k , t ) , z k ) , ( ϕ k ( z k , t ) , z k ) ∈ W k ∩ W i , (3.3.4) [Λ , w αi − ϕ αi ( z i , t )] | w i = ϕ i ( z i ,t ) = 0 , α = 1 , ..., r (3.3.5) EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 23
Recall Notation 1. Then the equalities (3.3.4) and (3.3.5) are equivalent to the system of congruences ϕ mi ( g ik ( ϕ mk ( z k , t ) , z k ) , t ) ≡ m f ik ( ϕ mk ( z k , t ) , z k ) , m = 1 , , , · · · (3.3.6) [Λ , w αi − ϕ αmi ( z i , t )] | w i = ϕ mi ( z i ,t ) ≡ m , m = 1 , , , · · · , α = 1 , ..., r. (3.3.7)We will construct the formal power series ϕ mi ( z i , t ) satisfying (3 . . m and (3 . . m by induction on m .We define ϕ αi | ( z i , t ) = P lρ =1 t ρ γ αρi ( z i ) , α = 1 , ..., r . Then from (3 . . ϕ i | ( z i , t ) satisfies (3 . . . On theother hand, since [Λ , γ αρi ( z i )] | w i =0 = P rβ =1 γ βρi ( z i ) T βiα ( z i , , α = 1 , ..., r from (3 . . , ϕ αi | ( z i , t )] | w i =0 = [Λ , l X ρ =1 t ρ γ αρi ( z i )] | w i =0 = l X ρ =1 r X β =1 t ρ γ βρi ( z i ) T βiα ( z i ,
0) = r X β =1 ϕ βi | ( z i , t ) T βiα ( z i , , ϕ αi | ( z i , t )] = P rβ =1 ϕ βi | ( z i , t ) T βiα ( w i , z i ) − P rβ =1 w βi P βiα ( w i , z i , t ) for some P βiα ( w i , z i , t )which are homogenous polynomials in t , ..., t l of degree 1 with coefficients in Γ( W i , T W ). Hence from(3 . . , w αi − ϕ αi | ( z i , t )] = r X β =1 ( w βi − ϕ βi | ( z i , t )) T βiα ( w i , z i ) + r X β =1 w βi P βiα ( w i , z i , t )so that we obtain [Λ , w αi − ϕ i | ( z i , t )] | w i − ϕ i | ( z i ,t ) ≡ ϕ i | ( z i , t ) satisfies (3 . . . Hence the inductionholds for m = 1.Now we assume that we have already constructed ϕ mi ( z i , t ) = ( ϕ mi ( z i , t ) , · · · , ϕ αmi ( z i , t ) , · · · , ϕ rmi ( z i , t ))satisfying (3 . . m and (3 . . m such that [Λ , w αi − ϕ αmi ( z i , t )] is of the form[Λ , w αi − ϕ αmi ( z i , t )] = r X β =1 ( w βi − ϕ βmi ( z i , t )) T βiα ( w i , z i ) + Q αmi ( z i , t ) + r X β =1 w βi P βmiα ( w i , z i , t )(3.3.8)such that the degree of P βmiα ( w i , z i , t ) is at least 1 in t , ..., t l . We note that we can rewrite (3.3.8) in thefollowing way. r X β =1 ( w βi − ϕ βmi ( z i , t )) T βiα ( w i , z i ) + Q αmi ( z i , t ) + r X β =1 φ βmi ( z i , t ) P βmiα ( ϕ mi ( z i , t ) , z i , t ) + r X β =1 ( w βi − ϕ βmi ( z i , t )) L βiα ( w i , z i , t )such that the degree of L βiα ( w i , z i , t ) in t , ..., t l is at least 1 so that (3.3.8) becomes the following form:[Λ , w αi − ϕ αmi ( z i , t )] = r X β =1 ( w βi − ϕ βmi ( z i , t )) T βiα ( w i , z i ) + K αmi ( z i , t ) + r X β =1 ( w βi − ϕ βmi ( z i , t )) L βiα ( w i , z i , t )(3.3.9)where K αmi ( z i , t ) := Q αmi ( z i , t ) + P rβ =1 φ βmi ( z i , t ) P βmiα ( ϕ mi ( z i , t ) , z i , t ).We set ψ ik ( z k , t ) := [ ϕ mi ( g ik ( ϕ mk ( z k , t ) , z k ) , t ) − f ik ( ϕ mk ( z k , t ) , z k )] m +1 (3.3.10) G αi ( z i , t ) := [[Λ , w αi − ϕ αmi ( z i , t )] | w i = ϕ mi ( z i ,t ) ] m +1 , α = 1 , ..., r. (3.3.11)We claim that { ( ψ ik ( z k , t ) , ..., ψ rik ( z k , t )) }⊕{ ( − G i ( z i , t ) , ..., − G ri ( z i , t )) } defines a 1-cocycle in the followingˇCech reolution of N • V/W : C ( U ∩ V, N W/V ⊗ ∧ T W | V ) ∇ x C ( U ∩ V, N W/V ⊗ T W | V ) δ −−−−→ C ( U ∩ V, N W/V ⊗ T W | V ) ∇ x ∇ x C ( U ∩ V, N W/V ) − δ −−−−→ C ( U ∩ V, N W/V ) δ −−−−→ C ( U ∩ V, N W/V ) By defining ψ ik ( z, t ) = ψ ik ( z k , t ) for (0 , z k ) ∈ U k ∩ U i , we have the equality (see [Kod62] p.152-153) ψ ik ( z, t ) = ψ ij ( z, t ) + F ij · ψ jk ( z, t ) , for z ∈ U i ∩ U j ∩ U k (3.3.12)On the other hand, by applying [Λ , − ] on (3 . . , [Λ , w αi − ϕ αmi ]] = r X β =1 − [Λ , w βi − ϕ βmi ( z i , t )] ∧ T βiα ( w i , z i ) + r X β =1 ( w βi − ϕ βmi ( z i , t ))[Λ , T βiα ( w i , z i )](3.3.13)+[Λ , K αmi ( z i , t )] + r X β =1 − [Λ , w βi − ϕ βi ( z i , t )] ∧ L βiα ( w i , z i , t ) + r X β =1 ( w βi − ϕ βi ( z i , t ))[Λ , L βiα ( w i , z i , t )]By restricting (3 . .
13) to w i = ϕ mi ( z i , t ), since G αi ( z i , t ) ≡ m , α = 1 , ..., r , we get0 ≡ m +1 r X β =1 − G βi ( z i , t ) ∧ T βiα (0 , z i ) + [Λ , K αmi ( z i , t )] | w i = ϕ mi ( z i ,t ) + r X β =1 − G βi ( z i , t ) ∧ L βiα ( ϕ mi ( z i , t ) , z i , t )Since the degree L ( w i , z i , t ) is at least 1 in t , ..., t l and we have, from (3 . . G αi ( z i , t ) ≡ m +1 K αmi ( z i , t ) = Q αmi ( z i , t ) + r X β =1 φ βmi ( z i , t ) P βmiα ( ϕ βmi ( z i , t ) , z i , t ) , (3.3.14)we obtain 0 ≡ m +1 r X β =1 − G βi ( z i , t ) ∧ T βiα (0 , z i ) + [Λ , G αi ( z i , t )] | w i =0 (3.3.15)Next, since f αik ( w k , z k ) − ϕ αmi ( g ik ( w k , z k ) , t ) − [ f αik ( ϕ mk ( z k , t ) , z k ) − ϕ αmi ( g ik ( ϕ mk ( z k , t ) , z k ) , t )] = P rβ =1 ( w βk − ϕ βmk ( z k , t )) · S βkα ( w k , z k , t ) for some S βkα ( w k , z k , t ). By setting t = 0, we get f αik ( w k , z k ) − f αik (0 , z k ) = P rβ =1 w βk · S βkα ( w k , z k , w γk and setting w k = 0, we obtain ∂f αik ( w k , z k ) ∂w γk | w k =0 = S γkα (0 , z k , . (3.3.16)Then we have[Λ , f αik ( w k , z k ) − ϕ αmi ( g ik ( w k , z k ) , t )] | w k = ϕ mk ( z k ,t ) + [Λ , ψ αik ( z k , t )] | w k =0 ≡ m +1 [Λ , f αik ( w k , z k ) − ϕ αmi ( g ik ( w k , z k ) , t )] | w k = ϕ mk ( z k ,t ) + [Λ , ψ αik ( z k , t )] | w k = ϕ mk ( z k ,t ) ≡ m +1 [Λ , f αik ( w k , z k ) − ϕ αmi ( g ik ( w k , z k ) , t ) + ψ αik ( z k , t )] | w k = ϕ mk ( z k ,t ) ≡ m +1 [Λ , r X β =1 ( w βk − ϕ βmk ( z k , t )) S βkα ( w k , z k , t )] | w k = ϕ mk ( z k ,t ) ≡ m +1 r X β =1 [Λ , w βk − ϕ βmk ( z k , t )] | w k = ϕ mk ( z k ,t ) · S βkα ( ϕ mk ( z k , t ) , z k , t ) ≡ m +1 r X β =1 G βk ( z k , t ) · S βkα (0 , z k , ≡ m +1 r X β =1 G βk ( z k , t ) · ∂f αik ( w k , z k ) ∂w βk | w k =0 Hence we obtain the equality[Λ , f αik ( w k , z k ) − ϕ αmi ( g ik ( w k , z k ) , t )] | w k = ϕ mk ( z k ,t ) + [Λ , ψ αik ( z k , t )] | w k =0 (3.3.17) ≡ m +1 r X β =1 G βk ( z k , t ) · S βkα (0 , z k ,
0) = r X β =1 G βk ( z k , t ) · ∂f αik ( w k , z k ) ∂w βk | w k =0EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 25 On the other hand, from (3.3.9), we have[Λ , f αik ( w k , z k ) − ϕ αmi ( g ik ( w k , z k ) , t )] = r X β =1 ( f βik ( w k , z k ) − ϕ βmi ( g ik ( w k , z k ) , t )) T βiα ( w i , z i )(3.3.18) + K αmi ( z i , t ) + r X β =1 ( f βik ( w k , z k ) − ϕ βmi ( g ik ( w k , z k ) , t )) L βiα ( w i , z i , t )By restricting (3.3.18) to w k = ϕ mk ( z k , t ), we get, from (3 . . , f αik ( w k , z k ) − ϕ αmi ( g ik ( w k , z k ) , t )] | w k = ϕ mk ( z k ,t ) ≡ m +1 r X β =1 − ψ βik ( z k , t ) T βiα (0 , z i ) + G αi ( z i , t )(3.3.19)Hence from (3.3.17), (3.3.19) and (3 . . r X β =1 − ψ βik ( z k , t ) T βiα (0 , z i ) + G αi ( z i , t ) + [Λ , ψ αik ( z k , t )] | w k =0 = r X β =1 G βk ( z k , t ) · ∂f αik ( w k , z k ) ∂w βk | w k =0 (3.3.20)Hence from (3.3.12),(3.3.17),(3.3.20), { ( ψ ik ( z k , t ) , ..., ψ rik ( z k , t )) } ⊕ { ( G i ( z i , t ) , ..., G ri ( z i , t )) } defines a 1-cocycle in the above complex so that we get the claim. We call ψ m +1 ( t ) := { ( ψ ik ( z k , t ) , ..., ψ rik ( z k , t )) } and G m +1 ( t ) := { ( G i ( z i , t ) , ..., G ri ( z i , t )) } the m -th obstruction so that the coefficients of ( ψ m +1 ( t ) , G m +1 ( t )) in t , ..., t l lies in H ( V, N • V/W ).On the other hand, by hypothesis, the cohomology group H ( V, N • V/W ) vanishes. Therefore there ex-ists ϕ αi | m +1 ( z i , t ) such that ψ ik ( z, t ) = F ik ( z ) ϕ k | m +1 ( z, t ) − ϕ i | m +1 ( z, t ) and P rβ =1 ϕ βi | m +1 ( z i , t ) T βiα (0 , z i ) − [Λ , ϕ αi | m +1 ( z i , t )] | w i =0 = − G αi ( z i , t ). Then we can show (3 . . m +1 (for the detail, see [Kod62] p.154). Onthe other hand, − [Λ , ϕ αi | m +1 ( z i , t )] = − r X β =1 ϕ βi | m +1 ( z i , t ) T βiα ( w i , z i ) − G αi ( z i , t ) + r X β =1 w βi R βiα ( w i , z i , t )where the degree of R βiα ( w i , z i , t ) is m + 1 in t , ..., t l . Then from (3 . . , w αi − ϕ αmi ( z i , t ) − ϕ αi | m +1 ( z i , t )] = [Λ , w αi − ϕ αmi ( z i , t )] + [Λ , − ϕ αi | m +1 ( z i , t )](3.3.21)= r X β =1 ( w βi − ϕ βmi ( z i , t ) − ϕ βi | m +1 ( z i , t )) T βiα ( w i , z i ) + Q αmi ( z i , t ) − G αi ( z i , t ) + r X β =1 w αi ( P βmiα ( w i , z i , t ) + R βiα ( w i , z i , t ))By setting ϕ α ( m +1) i ( z i , t ) := ϕ αmi ( z i , t )+ ϕ αi | m +1 ( z i , t ), we show that [Λ , w αi − ϕ α ( m +1) i ( z i , t )] | w i = ϕ m +1 i ( z i ,t ) ≡ m +1
0. Indeed, from (3 . .
21) and (3 . . , w αi − ϕ α ( m +1) i ( z i , t )] | w i = ϕ m +1 i ( z i ,t ) ≡ m +1 Q αmi ( z i , t ) − G αi ( z i , t ) + r X β =1 ( ϕ αmi ( z i , t ) + ϕ i | m +1 ( z i , t )) P βmiα ( ϕ mi ( z i , t ) + ϕ i | m +1 ( z i , t ) , z i , t ) ≡ m +1 Q αmi ( z i , t ) − G αi ( z i , t ) + r X β =1 ϕ αmi ( z i , t ) P βmiα ( ϕ mi ( z i , t ) , z i , t ) ≡ m +1 K αmi ( z i , t ) − G αi ( z i , t ) ≡ m +1 . . m +1 . This completes the inductive construction of the polynomials ϕ mi ( z i , t ) , i ∈ I .3.4. Proof of convergence.
We will show that we can choose ϕ i | m ( z i , t ) in each inductive step so that the formal power series ϕ i ( z i , t ) , i ∈ I constructed in the previous subsection, converges absolutely for | t | < ǫ for a sufficientlysmall number ǫ > Notation 3.
Recall Notation 2. We write A ( t ) = a b ∞ X n =1 b n ( t + · · · + t l ) n n (3.4.1) instead A ( t ) in Notation 2 to keep the notational consistency with [Kod62] . We also note that A ( t ) v ≪ (cid:16) ab (cid:17) v − A ( t ) for v = 2 , , .... (3.4.2)We may assume that | F λikµ (0 , z ) | < c with c >
1. Then ϕ i | ( z i , t ) ≪ A ( t ) if b is sufficiently large.We assume that ϕ mi ( z i , t ) ≪ A ( t )(3.4.3)for an integer m ≥
1, we shall estimate the coefficients of the homogenous polynomials ψ ik ( z, t ) and G αi ( z, t )from (3 . .
10) and (3 . . W δi be the subdomain of W i consisting of all points ( w i , z i ), | w i | < − δ, | z i | < − δ for a sufficientlysmall number δ > { W δi | i ∈ I } forms a covering of W , and { U δi = W δi ∩ V | i ∈ I } forms a coveringof V .First we estimate the coefficients of the homogeneous polynomials ψ ik ( z, t ). We briefly summarize Ko-daira’s result in the following: we expand f ik ( w k ) = f ik ( w k , z k ) and g ik ( w k ) = g ik ( w k , z k ) into power seriesin w k , ..., w rk whose coefficients are vector-valued holomorphic functions of z = (0 , z k ) defined on U k ∩ U i .We assume that f ik ( w k ) ≪ P ∞ n =1 c n ( w k + · · · + w rk ) n and g ik ( w k ) = P ∞ n =0 c n ( w k + · · · + w rk ) n . Then we canestimate ψ ik ( z k , t ) ≪ c A ( t ) , z k ∈ U k ∩ U i , (3.4.4)where c = 2 rc c rab (cid:16) d δ + rc (cid:17) with b > max { c ra, c raδ } . (3.4.5)Second we estimate the coefficients of the homogeneous polynomials G αi ( z, t ) , α = 1 , ..., r . Let Λ =Λ i ( w i , z i ) = P r + dp,q =1 Λ ipq ( w i , z i ) ∂∂x pi ∧ ∂∂x qi with Λ ipq ( w i , z i ) = − Λ ipq ( w i , z i ), where x i = ( w i , z i ) on W i . By con-sidering coefficients Λ ipq ( w i , z i ) of ∂∂x pi ∧ ∂∂x qi , we can consider Λ i ( w i , z i ) a vector-valued holomorphic functionon W i . We expand Λ i ( w i ) = Λ i ( w i , z i ) into power series in w i , ..., w ri whose coefficients are vector valuedholomorphic functions of z = (0 , z i ) defined on U i and we may assume, for any p, q ,Λ ipq ( w i ) = Λ ipq ( w i , z i ) ≪ ∞ X n =0 e n ( w i + · · · + w ri ) n (3.4.6)for some constant e >
0. Now we estimate G αi ( z i , t ) = [[Λ , w αi − ϕ αmi ( z i , t )] | w i = ϕ mi ( z i ,t ) ] m +1 for z i ∈ U δi . (3.4.7)First we estimate [[Λ , w αi ] | w i = ϕ mi ( z i ,t ) ] m +1 in (3 . . , w αi ] | w i = ϕ mi ( z i ,t ) = d + r X p,q =1 ipq ( ϕ mi ( z i , t ) , z i ) ∂w αi ∂x pi ∂∂x qi (3.4.8)Since constant terms and linear terms of Λ ipq ( w i , z i ) with respect to w i , · · · , w ri does not contribute to[Λ ipq ( ϕ mi ( z i , t ) , z i )] m +1 , we get, from (3 . .
6) and (3 . . ipq ( ϕ mi ( z i , t ) , z i )] m +1 ≪ ∞ X n =2 e n r n A ( t ) n = e r ∞ X n =1 e n r n A ( t ) n +1 ≪ e rA ( t ) ∞ X n =1 (cid:16) e rab (cid:17) n = e r ab A ( t ) ∞ X n =0 (cid:16) e rab (cid:17) n (3.4.9)Assuming that b > e ra, (3.4.10) EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 27 we obtain, from (3 . . . . , w αi ] | w i = ϕ mi ( z i ,t ) ] m +1 ≪ d + r ) e r ab A ( t )(3.4.11)Second, we estimate [[Λ , ϕ αmi ( z i , t )] | w i = ϕ mi ( z i ,t ) ] m +1 in (3 . . . . ∂ϕ αmi ( z i , t ) ∂w βi = 0 , ϕ αmi ( z i , t ) ∂z γi = 12 πi Z | ξ − z γi | = δ ϕ αmi ( z i , · · · , ξ, · · · , z di , t )( ξ − z γi ) dξ ≪ A ( t ) δ for | z i | < − δ. (3.4.12)Since constant term of Λ ipq ( w i , z i ) with respect to w i , ..., w ri does not contribute to [[Λ , ϕ αmi ( z i , t )] | w i = ϕ mi ( z i ,t ) ] m +1 ,we get, from (3 . . . .
2) and (3 . . , ϕ αmi ( z i , t )] | w i = ϕ αmi ( z i ,t ) ] m +1 = r + d X p,q =1 ipq ( ϕ αmi ( z i , t ) , z i ) ∂ϕ αmi ( z i , t ) ∂x pi ∂∂x qi (3.4.13) ≪ r + d ) A ( t ) δ ∞ X n =1 e n r n A ( t ) n , z i ∈ U δi ≪ r + d ) δ ∞ X n =1 e n r n A ( t ) n +1 ≪ r + d ) δ ∞ X n =1 (cid:16) e rab (cid:17) n A ( t ) ≪ r + d ) δ (cid:16) e rab (cid:17) A ( t ) ∞ X n =0 (cid:16) e rab (cid:17) n Assuming that b > e ra , we get[[Λ , ϕ αmi ( z i , t )] | w i = ϕ αmi ] m +1 ≪ r + d ) e raδb A ( t ) , z i ∈ U δi . (3.4.14)Hence from (3.4.11), and (3.4.14), we obtain G αi ( z i , t ) = [[Λ , w αi − ϕ mαi ( z i , t )] | w i = ϕ mi ( z i ,t ) ] m +1 ≪ e A ( t ) , z i ∈ U δi , (3.4.15)where e = 4( d + r ) e r ab + 4( r + d ) e raδb . (3.4.16) Lemma 3.4.17.
We can choose the homogenous polynomials ϕ i | m +1 ( z, t ) , i ∈ I satisfying ψ ik ( z, k ) = F ik ( z ) ϕ k | m +1 ( z, t ) − ϕ i | m +1 ( z, t ) − G αi ( z, t ) = − [ ϕ αi | m +1 ( z, t ) , Λ ] | w i =0 + r X β =1 ϕ βi | m +1 ( z, t ) T βiα (0 , z ) in such a way that ϕ i | m +1 ( z, t ) ≪ c ( e + c ) A ( t ) , where c is independent of m .Proof. For any 0-cochain ϕ = { ϕ i ( z ) } , 1-cochain ( ψ, G ) = ( { ψ ik ( z ) } , { P rα =1 G αi ( z ) e αi } ), we define the normsof ϕ and ( ψ, G ) by || ϕ || := max i sup z ∈ U i | ϕ i ( z ) | , || ( ψ, G ) || := max i,k sup z ∈ U i ∩ U k | ψ ik ( z ) | + max i,α sup z ∈ U δi | G αi ( z ) | The coboundary ϕ is defined by˜ δ ( ϕ ) := ( − F ik ( z ) ϕ k ( z, t ) + ϕ i ( z ) , − [ ϕ αi ( z i ) , Λ ] | w i =0 + r X β =1 ϕ βi ( z ) T βiα (0 , z i ))For any coboundary ( ψ, G ), we define ι ( ψ ) = inf ˜ δϕ =( ψ,G ) || ϕ || . To prove Lemma 3.4.17, it suffices to show the existence of a constant c such that ι ( ψ, G ) ≤ c || ( ψ, G ) || . As-sume that such a constant c does not exist. Then we can find a sequence ( ψ ′ , G ′ ) , ( ψ ′′ , G ′′ ) , · · · , ( ψ ( µ ) , G ( µ ) ) , · · · such that there exists ϕ ( µ ) with δϕ ( µ ) = ( ψ ( µ ) , G ( µ ) ) satisfying || ϕ ( µ ) || <
2. Then we can show thatthere is a subsequence ϕ µ i , ϕ µ i , · · · such that ϕ ( µ v ) i ( z i ) converges absolutely and uniformly on U i . Let ϕ i ( z i ) = lim v ϕ ( µ v ) i ( z i ) and let ϕ = { ϕ i ( z i ) } . Then we have || ϕ µ v − ϕ || →
0. On the other hand˜ δ ( ϕ ) = (0 , G ϕ ), where G ϕ ( z ) = 0 for z ∈ U δi . By identity theorem G ϕ ( z ) = 0 for z ∈ U i so that ˜ δ ( ϕ ) = (0 , δ ( ϕ ( µ v ) − ϕ ) = ( ϕ ( µ v ) , G ( µ v ) ) which contradict to ι ( ϕ ( µ v ) ) = 1. (cid:3) From (3 . .
4) and (3 . . c ( c + e ) = c c + c e = 8 c c c r ab (cid:18) d δ + rc (cid:19) + c (cid:18) d + r ) e r ab + 4( r + d ) e raδb (cid:19) (3.4.18)From (3 . . , (3 . . . .
18) and Lemma 3.4.17, by assuming b > max { c c c r a (cid:18) d δ + rc (cid:19) + c (cid:18) d + r ) e r a + 4( r + d ) e raδ (cid:19) , c ra, c raδ , e ra } we can choose ϕ i | m +1 ( z i , t ) ≪ A ( t ) and so ϕ i ( z i , t ) ≪ A ( t ) so that the power series ϕ i ( z i , t ) converges for | t | < lb . Then by the argument of [Kod62] p.158, we obtain the equality ϕ i ( g ik ( ϕ k ( z k , t ) , z k ) , t ) = f ik ( ϕ k ( z k , t ) , z k ) , for | t | < ǫ, ( ϕ k ( z k , t ) , z k ) ∈ W δi ∩ W δk [Λ , w αi − ϕ αi ( z i , t )] | w i = ϕ i ( z i ,t ) = 0for a sufficiently small number ǫ >
0, which proves Theorem 3.3.1. (cid:3)
In the case H ( V, N • V/W ) = 0, our proof of Theorem 3 . . Theorem 3.4.19.
If the obstruction ( ψ m +1 ( t ) , G m +1 ( t )) vanishes for each integer m ≥ , then there existsa Poisson analytic family V of compact holomorphic Poisson submanifolds V t , t ∈ M , of ( W, Λ ) such that V = V and the characteristic map σ : T ( M ) → H ( V, N • V/W ) ∂∂t ( ∂V t ∂t ) t =0 is an isomorphism. Maximal families: Theorem of completeness.
We note that Definition 2.4.1 can be extended to arbitrary codimensions.
Theorem 3.5.1 (theorem of completeness) . Let V be a Poisson analytic family of compact holomorphicPoisson submanifolds V t , t ∈ M , of ( W, Λ ) . If the characteristic map σ : T ( M ) → H ( V , N • V /W ) ∂∂t (cid:18) ∂V t ∂t (cid:19) t =0 is an isomorphism, then the family V is maximal at t = 0 .Proof. We extend the arguments in [Kod62] p.158-160 in the context of holomorphic Poisson deformations.Consider an arbitrary Poisson analytic family V ′ of compact holomorphic Poisson submanifolds V ′ s , s ∈ M ′ of ( W, Λ ) such that V ′ = V , where M ′ = { s = ( s , ..., s q ) ∈ C q || s | < } . We shall construct a holomorphicmap h : s → t = h ( s ) of a neighborhood N ′ of 0 into M such that h (0) = 0 and V ′ s = V h ( s ) .We keep the notations in 3.2 so that the holomorphic Poisson submanifold V t is defined in each domain W i , i ∈ I by w i = ϕ i ( z i , t ) and satisfy[Λ , w αi − ϕ αi ( z i , t )] = r X β =1 ( w βi − ϕ βi ( z i , t )) T βiα ( w i , z i , t ) . (3.5.2) EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 29
We may assume that V ′ s is defined in each domain W i , i ∈ I , by w i = θ i ( z i , t ), where θ i ( z i , t ) is a vector-valuedholomorphic function of z i , s with | z i | < , | s | <
1, and satisfy[Λ , w αi − θ αi ( z i , s )] = r X β =1 ( w βi − θ βi ( z i , s )) P βiα ( w i , z i , s )(3.5.3)for some P βiα ( w i , z i , s ) which are power series in s with coefficients in Γ( W i , T W ) and P βiα (0 , z i ,
0) = T βiα (0 , z i ).Then V ′ s = V h ( s ) is equivalent to θ i ( z i , s ) = ϕ i ( z i , h ( s ))(3.5.4)Recall Notation 1 and let us write h ( s ) = h ( s ) + h ( s ) + · · · , ϕ i ( z i , t ) = ϕ i | ( z i , t ) + ϕ i | ( z i , t ) + · · · , and θ i ( z i , s ) = θ i | ( z i , t ) + θ i | ( z i , t ) + · · · . We will construct h ( s ) satisfying (3 . .
4) by solving the system ofcongruences by induction on mθ i ( z i , s ) ≡ m ϕ i ( z i , h m ( s )) , i ∈ I, m = 1 , , , ... (3.5.5)Since σ : T ( M ) → H ( V , N • V /W ) is an isomorphism by hypothesis, { ∂ϕ i ( z i ,t ) ∂t | t =0 , ...., ∂ϕ i ( z i ,t ) ∂t l | t =0 } is abasis of H ( V , N • V /W ). Since { θ i | ( z i , s ) } of θ i | ( z i , s ) , i ∈ I , represents a linear form in s whose coefficientsare in H ( V , N • V /W ), there exists a linear vector-valued function h ( s ) such that θ i | ( z i , s ) = ϕ i | ( z i , h ( s ))which proves (3 . . . Now assume that we have already constructed h m ( s ) satisfying (3 . . m . Wewill find h m +1 ( s ) such that h m +1 ( s ) = h m ( s ) + h m +1 ( s ) satisfy (3 . . m +1 . Let ω i ( z i , s ) = [ θ i ( z i , s ) − ϕ i ( z i , h m ( s ))] m +1 . We claim that ω i ( z i , s ) = F ik ( z ) · ω k ( z k , s ) , z ∈ U i ∩ U k (3.5.6) − [ ω αi ( z i , s ) , Λ ] | w i =0 + r X β =1 ω βi ( z i , s ) T βiα (0 , z i ) = 0 , (3.5.7)For the proof of (3 . . . . . .
2) and (3 . . , ω αi ( z i , s )] | w i =0 ≡ m +1 [Λ , ω αi ( z i , s )] | w i = θ i ( z i ,t ) ≡ m +1 [Λ , θ αi ( z i , s ) − w αi + w αi − ϕ αi ( z i , h m ( s ))] | w i = θ i ( z i ,t ) ≡ m +1 − [Λ , w αi − θ αi ( z i , s )] | w i = θ i ( z i ,t ) + [Λ , w αi − ϕ αi ( z i , h m ( s ))] | w i = θ i ( z i ,t ) ≡ m +1 [Λ , w αi − ϕ αi ( z i , h m ( s ))] | w i = θ i ( z i ,t ) ≡ m +1 r X β =1 ( θ βi ( z i , s ) − ϕ βi ( z i , h ( s ))) T βiα ( w i , z i , h ( s )) ≡ m +1 r X β =1 ω βi ( z i , s ) T βiα (0 , z i )This proves (3 . . . .
6) and (3 . . { ω i ( z i , s ) } is a homogenous polynomial of degree m +1 in s withcoefficents in H ( V, N • V /W ) so that there exists a homogenous polynomial h m +1 ( s ) of degree 1 in s such that ω i ( z i , s ) = ϕ i | ( z i , h m +1 ( s )). Therefore we have ϕ i ( z i , h m +1 ( s )) ≡ m +1 ϕ i ( z i , h m ( s )) + ω i ( z i , s ) ≡ m +1 θ i ( z i , s )which completes the inductive construction of h m +1 ( s ) satisfying (3 . . m +1 .3.6. Proof of convergence.
The convergence of the power series h ( s ) follows from the same arguments in [Kod62] p.160-161. Thiscompletes the proof of Theorem 3.5.1. (cid:3) Example 3.
Let [ ξ , ξ , ξ , ξ ] be the homogenous coordinate on P C and a hyperplane V defined by ξ = 0 so that N V/ P C ∼ = O V (1) . Let [1 , z , z , z ] = [1 , ξ ξ , ξ ξ , ξ ξ ] . Consider a Poisson structure Λ = z ∂∂z ∧ ∂∂z on P C . Then V ∼ = P C is a holomorphic Poisson submanifold. We compute H ( V, N • V/ P C ) which is the kernel ∇ : H ( V, O V (1)) → H ( V, T P C | V (1)) . Since [Λ , z ] = 0 , ∇ ( az + bz + c ) = − [Λ , az + bz + c ] | z =0 = − az ∂∂z + bz ∂∂z = 0 ⇐⇒ a = b = 0 so that dim C H ( V, O V (1)) = 1 . Since [Λ , z − t ] | z = t = 0 , holomorphic Poisson deformations of V in ( P C , Λ ) is unobstructed, and explicitly we have a Poisson analytic family of holomorphic Poisson submani-folds V ⊂ ( P C × C , Λ ) defined by ξ − tξ = 0 whose characteristic map T C → H ( V, N • V/ P C ) = H ( P C , O P C (1) • ) a a = 0 · z + 0 · z + a is an isomorphism so that V is complete. Example 4.
Let us consider the Poisson structure Λ = z ∂∂z ∧ ∂∂z on P C as in Example 3 and a holo-morphic Poisson submanifold V defined by ξ = ξ = 0 which is a nonsingular rational curve ∼ = P C and thenormal bundle N V/ P C is isomorphic to O P C (1) ⊕ O P C (1) . We compute H ( V, N • V/ P C ) which is the kernel of ∇ : H ( P C , O P C (1) ⊕ O P C (1)) → H ( P C , T P C | P C (1) ⊕ T P C | P C (1)) . Since [Λ , z ] = z ∂∂z , and [Λ , z ] = 0 , ∇ (( az + b ) ⊕ ( cz + d )) = ( − [Λ , az + b ] | z = z =0 + ( az + b ) ∂∂z ) ⊕ ( − [Λ , cz + d ] | z = z =0 ) = 0 ⇐⇒ a = b = 0 . so that dim C H ( V, N • V/ P C ) = 2 . Since [Λ , z ] | z = z − t z − t =0 = 0 and [Λ , z − t z − t ] | z = z − t z − t = 0 ,holomorphic Poisson deformations of V in ( P C , Λ ) is unobstructed, and explicitly we have a Poisson analyticfamily of holomorphic Poisson submanifolds V ⊂ ( P C × C , Λ )) defined by ξ = ξ − t ξ − t ξ = 0 whosecharacteristic map is T C → H ( V, N • V/ P C ) = H ( P C , ( O P C (1) ⊕ O P C (1)) • )( a , a ) (0 , a z + a ) which is an isomorphism so that V is complete. Example 5.
Let ( X, Λ ) be a non-degenerate Poisson K surface. Consider ( X × C q , Λ ) and let ( w , ..., w q ) be the coordinate of C q . Then w = · · · = w q = 0 defines a holomorphic Poisson submanifold which is ( X, Λ ) and the normal bundle N X/X × C q is ⊕ q O X . Since Λ | ( x,a ) ∈ ∧ T X | x ⊂ ∧ T X × C | ( x,a ) for each ( x, a ) ∈ X × C q ,and [Λ , w i ] = 0 for i = 1 , ..., q , the obstructions lies in the first cohomology group of the following complexof sheaves ⊕ q O • X : ⊕ q O X − [ − , Λ ] −−−−−→ ⊕ q T X − [ − , Λ ] −−−−−→ ⊕ q ∧ T X − [ − Λ ] −−−−−→ · · · whose the -th cohomology group is isomorphic to ⊕ q H ( X, C ) ∼ = C q and the first cohomology group isisomorphic to ⊕ q H ( X, C ) = 0 so that deformations of X in ( X × C q , Λ ) is unobstructed. Explicitly wehave a Poisson analytic family of holomorphic Poisson submanifolds V ⊂ (( X × C q ) × C q , Λ ) with ( t , ..., t q ) the last coordinate of C q which is defined by w − t = · · · = w q − t q = 0 whose characteristic map T C q → H ( X, N • X/X × C ) ∼ = ⊕ q H ( X, C ) ∼ = ⊕ q C ( a , ..., a q ) ( a , ..., a q ) is an isomorphism so that V is complete. Simultaneous deformations of holomorphic Poisson structures and compactholomorphic Poisson submanifolds
We extend the definition of a Poisson analytic family of compact holomorphic Poisson submanifolds inDefinition 3.0.1 by deforming holomorphic Poisson structures as well on a fixed complex manifold W . Definition 4.0.1.
Let W be a complex manifold of dimension d + r . We denote a point in W by w anda local coordinate of w by ( w , ..., w r + d ) . By an extended Poisson analytic family of compact holomorphicPoisson submanifolds of dimension d of W , we mean a holomorphic Poisson submanifold V ⊂ ( W × M, Λ) of codimension r , where M is a complex manifold and Λ is a holomorphic Poisson structure on W × M ,such that EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 31 (1) the canonical projection π : ( W × M, Λ) → M is a holomorphic Poisson fibre manifold as in Defi-nition . . so that Λ ∈ H ( W × M, ∧ T W × M/M ) and π − ( t ) := ( W, Λ t ) is a holomorphic Poissonsubmanifold of ( W × M, Λ) for each point t ∈ M . (2) for each point t ∈ M , V t × t := ω − ( t ) = V ∩ π − ( t ) is a connected compact holomorphic Poissonsubmanifold of ( W, Λ t ) of dimension d , where ω : V → M is the map induced from π . (3) for each point p ∈ V , there exist r holomorphic functions f α ( w, t ) , α = 1 , ..., r defined on a neighbor-hood U p of p in W × M such that rank ∂ ( f ,...,f r ) ∂ ( w ,...w r + d ) = r , and U p ∩ V is defined by the simultaneousequations f α ( w, t ) = 0 , α = 1 , ..., r .We call V ⊂ ( W × M, Λ) an extended Poisson analytic family of compact holomorphic Poisson submanifolds V t , t ∈ M of ( W, Λ t ) . We also call V ⊂ ( W × M, Λ) an extended Poisson analytic family of simultaneousdeformations of a holomorphic Poisson submanifold V t of ( W, Λ t ) for each fixed point t ∈ M . The extended complex associated with the normal bundle of a holomorphic Poisson sub-manifold of a holomorphic Poisson manifold.
Let V be a holomorphic Poisson submanifold of a holomorphic Poisson manifold ( W, Λ ). We will describea complex of sheaves to control simultaneous deformations of holomorphic Poisson structures and holomor-phic Poisson submanifolds. We recall that the complex associated with the normal bundle (see Definition3.1.13) N • V/W : N V/W ∇ −→ N V/W ⊗ T W | V ∇ −→ N V/W ⊗ ∧ T W | V ∇ −→ · · · (4.1.1)controls holomorphic Poisson deformations of V in ( W, Λ ), and the complex ∧ T • W : ∧ T W − [ − , Λ ] −−−−−→ ∧ T W − [ − , Λ ] −−−−−→ ∧ T W − [ − , Λ ] −−−−−→ · · · (4.1.2)controls deformations of the holomorphic Poisson structure Λ on the fixed underlying complex manifold W (see Appendix A). By combining two complexes (4 . .
1) and (4 . . W :( ∧ T W ⊕ i ∗ N V/W ) • : ∧ T W ⊕ i ∗ N V/W ˜ ∇ −→ ∧ T W ⊕ i ∗ ( N V/W ⊗ T W | V ) ˜ ∇ −→ ∧ T W ⊕ i ∗ ( N V/W ⊗ ∧ T W | V ) ˜ ∇ −→ · · · which controls simultaneous deformations of the holomorphic Poisson structure Λ and the holomorphicPoisson submanifold V of ( W, Λ ), where i : V ֒ → W is the embedding. We keep the notations in subsection3.1.We note that Γ( W i , ∧ p +2 T Y ⊕ i ∗ ( N V/W ⊗ ∧ p T W | V )) = Γ( W i , ∧ p +2 T Y ) ⊕ Γ( U i , N V/W ⊗ ∧ p T W | V ) ∼ =Γ( W i , ∧ p +2 T W ) ⊕ ( ⊕ r Γ( U i , ∧ p T W | V )). From these isomorphisms, we define˜ ∇ : ∧ p +2 T W ⊕ i ∗ ( N V/W ⊗ ∧ p T W | V ) → ∧ p +3 T W ⊕ i ∗ ( N V/W ⊗ ∧ p +1 T W | V )locally in the following way:Γ( W i , ∧ p +2 T W ) ⊕ ( ⊕ r Γ( U i , ∧ p T W | V )) ˜ ∇ −→ Γ( W i , ∧ p +3 T W ) ⊕ ( ⊕ r Γ( U i , ∧ p +1 T W | V )(Π i , r X α =1 g αi e αi ) ( − [Π i , Λ ] , r X α =1 [Π i , w αi ] | w i =0 e αi + ∇ ( r X α =1 g αi e αi ))In other words, (Π i , ( g i , ..., g ri )) ( − [Π i , Λ ] , [Π i , w i ] | wi =0 − [ g i , Λ ] | wi =0 + ( − p r X β =1 g βi ∧ T βi (0 , z i ) , · · · , [Π i , w ri ] | wi =0 − [ g ri , Λ ] | wi =0 + ( − p r X β =1 g βi ∧ T βir (0 , z i ) ) First we show that ˜ ∇ defines a complex, i.e ˜ ∇ ◦ ˜ ∇ = 0. Since ∇ ◦ ∇ = 0, we have˜ ∇ ( ˜ ∇ (Π i , r X α =1 g αi e αi )) = (0 , r X α =1 − [[Π i , Λ ] , w αi ] | w i =0 e αi + ∇ ( r X α =1 [Π i , w αi ] | w i =0 e αi ))= (0 , r X α =1 − [[Π i , Λ ] , w αi ] | w i =0 e αi + r X α =1 − [[Π i , w αi ] , Λ ] | w i =0 e αi + ( − p +1 r X α,β =1 [Π i , w αi ] | w i =0 ∧ T αiβ (0 , z i ) e βi ) Hence ˜ ∇ ◦ ˜ ∇ = 0 is equivalent to − [[Π i , Λ ] , w αi ] | w i =0 − [[Π i , w αi ] , Λ ] | w i =0 + ( − p +1 r X β =1 [Π i , w βi ] | w i =0 ∧ T βiα (0 , z i ) = 0(4.1.3)Let us show (4.1.3). We note that from (3 . . i , [Λ , w αi ]] = r X β =1 [Π i , w βi T βiα ( w i , z i )] = r X β =1 w βi [Π i , T βiα ( w i , z i )] + ( − p +1 r X β =1 [Π i , w βi ] ∧ T βiα ( w i , z i )= ⇒ [Π i , [Λ , w αi ]] | w i =0 = ( − p +1 r X β =1 [Π i , w βi ] | w i =0 ∧ T βiα (0 , z i )Then (4 . .
3) is equivalent to − [[Π i , Λ ] , w αi ] | w i =0 − [[Π i , w αi ] , Λ ] w i =0 + [Π i , [Λ , w αi ]] w i =0 = 0(4.1.4)which follows from − [[Π i , Λ ] , w αi ] − [[Π i , w αi ] , Λ ] + [Π i , [Λ , w αi ]] = 0 by the graded Jacobi identity. Thisproves ˜ ∇ ◦ ˜ ∇ = 0.Next we show that ˜ ∇ is well-defined. In other words, on U i ∩ U k , the following diagram commutes(4.1.5) Γ( W k , ∧ p +2 T W ) ⊕ ( ⊕ r Γ( U k , ∧ p T W | V )) on U i ∩ U k −−−−−−−→ ∼ = Γ( W i , ∧ p +2 T W ) ⊕ ( ⊕ r Γ( U i , ∧ p T W | V )) ˜ ∇ y y ˜ ∇ Γ( W k , ∧ p +3 T X ) ⊕ ( ⊕ r Γ( U k , ∧ p +1 T W | V )) on U i ∩ U k −−−−−−−→ ∼ = Γ( W i , ∧ p +3 T W ) ⊕ ( ⊕ r Γ( U i , ∧ p +1 T W | V ))Let (Π , P rα =1 g αk e αk ) ∈ Γ( W k , ∧ p +2 T W ) ⊕ ( ⊕ r Γ( U k , ∧ p T W | V )) and let ∇ ( P rα =1 g αk e αk ) = P rα =1 G αk e αk , where G αk = − [ g αk , Λ ] | w k =0 + ( − p P rβ =1 g βk ∧ T βkα (0 , z k ) ∈ Γ( U k , ∧ p +1 T W | V ).Then ˜ ∇ ((Π , P rα =1 g αk e αk )) = ( − [Π , Λ ] , P rα =1 [Π , w αk ] | w k =0 e αk + P rα =1 G αk e αk ) is identified on U i ∩ U k with( − [Π , Λ ] , r X β =1 r X α =1 F βikα (0 , z k )[Π , w αk ] | w k =0 ! e βi + r X β =1 ( r X α =1 F βikα (0 , z k ) G αk ) e βi )(4.1.6)On the other hand, (Π , P rα =1 g αk e αk ) is identified on U i ∩ U k with (Π , P rβ =1 (cid:16)P rα =1 F βikα (0 , z k ) g αk (cid:17) e βi ) ∈ Γ( W i , ∧ p +2 T W ) ⊕ ( ⊕ r Γ( U i , ∧ p T W | V )). Then we have˜ ∇ ((Π , r X β =1 r X α =1 F βikα (0 , z k ) g αk ! e βi ) = ( − [Π , Λ ] , r X β =1 [Π , w βi ] | w i =0 e βi + ∇ ( r X α,β =1 F βikα (0 , z k ) g αk ) e βi ))(4.1.7)Hence in order for the diagram (4.1.5) to commute, we have to show that (4 . .
6) coincides with (4 . . . .
11) and (3.1.12), it is enough to show [Π , w βi ] | w i =0 = P rα =1 F βαik (0 , z k )[Π , w αk ] | w k =0 whichcomes from (3.1.1), and w βi = r X α =1 F βikα ( w k , z k ) w αk = ⇒ [Π , w βi ] = r X α =1 ( F βikα ( w k , z k )[Π , w αk ] + w αk [Π , F βikα ]) . Hence ˜ ∇ is well-defined. Definition 4.1.8.
We call the complex defined as above ( ∧ T W ⊕ i ∗ N V/W ) • : ∧ T W ⊕ i ∗ N V/W ˜ ∇ −→ ∧ T W ⊕ i ∗ ( N V/W ⊗ T W | V ) → ∧ T W ⊕ i ∗ ( N V/W ⊗ ∧ T W | V ) → · · · the extended complex associated with the normal bundle N V/W of a holomorphic Poisson submanifold V of aholomorphic Poisson manifold W and denote its i -th hypercohomology group by H i ( W, ( ∧ T W ⊕ i ∗ N V/W ) • ) . EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 33
Infinitesimal deformations.
Let M = { t = ( t , ..., t l ) ∈ C l || t | < } . Consider an extended Poisson analytic family V ⊂ ( W × M , Λ) ofcompact holomorphic Poisson submanifolds V t , t ∈ M of ( W, Λ t ) and let V = V as in Definition 4.0.1. Wekeep the notations in subsection 3.2 and subsection 4.1 so that for | t | < ǫ for a sufficiently small number ǫ > V t is defined by w i = ϕ i ( z i , t ) on each neighborhood W i and, by setting w λti = w λi − ϕ λi ( z i , t ) , λ = 1 , ..., r , F tik ( z t ) := (cid:16) ∂w λti ∂w µtk ( z t ) (cid:17) λ,µ =1 ,...,r for z t ∈ V t ∩ W i ∩ W k defines the normal bundle N V t /W of V t in W . For anarbitrary tangent vector ∂∂t = P ρ γ ρ ∂∂t ρ of M at t, | t | < ǫ , we let ψ i ( z t , t ) = ∂ϕ i ( z i ,t ) ∂t for z t = ( ϕ i ( z i , t ) , z i ).Then we obtain the equality ψ i ( z t , t ) = F tik ( z t ) · ψ k ( z t , t ) , for z t ∈ V t ∩ W i ∩ W k . (4.2.1)On the other hand, let Λ i ( w i , z i , t ) be the holomorphic Poisson structure Λ on W i × M . Then[Λ i ( w i , z i , t ) , Λ i ( w i , z i , t )] = 0(4.2.2)and Λ i ( w i , z i , t ) is of the formΛ i ( w i , z i , t ) := Λ i ( x i , t ) = r + d X α,β =1 Λ iαβ ( x i , t ) ∂∂x αi ∧ ∂∂x βi , with Λ iαβ ( x i , t ) = − Λ iβα ( x i , t ) , x i = ( w i , z i ) , by which we consider Λ i ( w i , z i , t ) as a vector-valued holomorphic function of ( w i , z i , t ). Let π i ( w i , z i , t ) = ∂ Λ i ( w i ,z i ,t ) ∂t . Then { π i ( w i , z i , t ) } ∈ H ( W, ∧ T W )(4.2.3)By taking the derivative of (4.2.2) with respect to t , we get[Λ i ( w i , z i , t ) , ∂ Λ i ( w i , z i , t ) ∂t ] = 0 ⇐⇒ − [ π i ( w i , z i , t ) , Λ i ( w i , z i , t )] = 0(4.2.4)Lastly, since w λi − ϕ λi ( z i , t ) = 0 , λ = 1 , ..., r define a holomorphic Poisson submanifold, we have[Λ i ( w i , z i , t ) , w λi − ϕ λi ( z i , t )] = r X µ =1 ( w µi − ϕ µi ( z i , t )) T µiλ ( w i , z i , t )(4.2.5)for some T µiλ ( w i , z i , t ) which is of the form T µiλ ( w i , z i , t ) = P µi ( w i , z i , t ) ∂∂w i + · · · + P µir ( w i , z i , t ) ∂∂w ri + Q µi ( w i , z i , t ) ∂∂z i + · · · + Q µid ( w i , z i , t ) ∂∂z di by which we consider T µiλ ( w i , z i , t ) as a vector valued holomorphic function of ( w i , z i , t ).By taking the derivative of (4.2.5) with respect to t , we get [ ∂ Λ i ( w i ,z i ,t ) ∂t , w λi − ϕ λi ( z i , t )]+[Λ i ( w i , z i , t ) , − ∂ϕ λi ( z i ,t ) ∂t ] = P rµ =1 − ∂ϕ µi ( z i ,t ) ∂t T µiλ ( w i , z i , t )+ P nµ =1 ( w λi − ϕ µi ( z i , t )) ∂T µiλ ( w i ,z i ,t ) ∂t . By restricting to V t , equivalently, by setting w i = ϕ i ( z i , t ), we get, on Γ( W i ∩ V t , T W | V t ),[ ∂ Λ i ( w i , z i , t ) ∂t , w λi − ϕ λi ( z i , t )] | V t − [Λ i ( w i , z i , t ) , ∂ϕ λi ( z i , t ) ∂t ] | V t = r X µ =1 − ∂ϕ µi ( z i , t ) ∂t T µiλ ( ϕ i ( z i , t ) , z i , t ) ⇐⇒ [ π i ( w i , z i , t ) , w λti ] | V t − [ ψ λi ( z t , t ) , Λ i ( w i , z i , t )] | V t + r X µ =1 ψ µi ( z t , t ) T µiλ ( ϕ i ( z i , t ) , z i , t ) = 0(4.2.6)Hence from (4.2.1), (4.2.3), (4.2.4) and (4.2.6), ( { π i ( w i , z i , t ) } , { ψ i ( z t , t ) } ) defines an element of H ( W, ( ∧ T W ⊕ i ∗ N V t /W ) • ) so that we have a linear map σ t : T t ( M ) → H ( W, ( ∧ T W ⊕ i ∗ N V t /W ) • ) ∂∂t ∂ (Λ t , V t ) ∂t := ( { π i ( w i , z i , t ) } , { ψ i ( z t , t ) } )We call σ t the characteristic map. Example 6.
Let [ ξ , ξ , ξ ] be the homogeneous coordinate on P C . Let [1 , z , z ] = [1 , ξ ξ , ξ ξ ] . Then V ⊂ ( P C × C , ( z + t z + t ) ∂∂z ∧ ∂∂z defined by ξ + t ξ + t ξ = 0 is an extended Poisson analytic family ofsimultaneous deformations of a holomorphic Poisson submanifold ξ = 0 of ( P C , Λ = z ∂∂z ∧ ∂∂z ) and wehave the characteristic map T C → H ( P C , ( ∧ T P C ⊕ i ∗ O P C (1)) • )( a, b ) a (cid:18) z ∂∂z ∧ ∂∂z , − z (cid:19) + b (cid:18) ∂∂z ∧ ∂∂z , − (cid:19) Example 7.
Let [ ξ , ξ , ξ , ξ ] be the homogenous coordinate on P C . Let [1 , z , z , z ] = [1 , ξ ξ , ξ ξ , ξ ξ ] . Then V ⊂ ( P C × C , ( z + t ) ∂∂z ∧ ∂∂z ) defined by ξ + t ξ = ξ + t ξ = 0 is an extended Poisson analytic familyof simultaneous deformations of a holomorphic Poisson submanifold ξ = ξ = 0 of ( P C , Λ = z ∂∂z ∧ ∂∂z ) and we have the characteristic map T C → H ( P C , ( ∧ T P C ⊕ i ∗ ( O P C (1) ⊕ O P C (1)) • )( a, b ) a ( ∂∂z ∧ ∂∂z , ( − , ⊕ b (0 , (0 , − Example 8.
We construct an extended Poisson analytic family of simultaneous deformations of holomor-phic Poisson submanifolds of a stable elliptic surface. As in [BHPVdV04] p. , let z ( s ) be an arbitraryholomorphic function on the unit disk ∆ = { s ∈ C || s | < } with Im z ( s ) > . Let G = Z × Z act on C × ∆ by ( m, n )( c, s ) = ( c + m + nz ( s ) , s ) . The quotient X = ( C × ∆) / ( Z × Z ) is a nonsingular surface fibered over ∆ such that X s is an elliptic curve with period , z ( s ) . Let c ′ = c + m + nz ( s ) , s ′ = s . Then we have ∂∂c = ∂∂c ′ ,and ∂∂s = nz ′ ( s ) ∂∂c ′ + ∂∂s ′ so that we get ∂∂c ∧ ∂∂s = ∂∂c ′ ∧ ∂∂s ′ and so Λ = ( s − t ) ∂∂c ∧ ∂∂s is a G -invariant bivectorfield on X × ∆ which defines a holomorphic Poisson structure Λ t on X for each t ∈ ∆ , and X t : s = t is aholomorphic Poisson submanifold of ( X, Λ t ) since the holomorphic Poisson structure Λ t degenerates along s = t . Then V ⊂ ( X × ∆ , Λ = ( s − t ) ∂∂c ∧ ∂∂s ) defined by s = t is an extended Poisson analytic family ofsimultaneous deformations of the holomorphic Poisson submanifold X : s = 0 of ( X, s ∂∂c ∧ ∂∂s ) , and we havethe characteristic map T C → H ( X, ( ∧ T X ⊕ i ∗ N X /X ) • ) a ( − a ∂∂c ∧ ∂∂s , a )4.3. Theorem of existence.Theorem 4.3.1 (theorem of existence) . Let V be a holomorphic Poisson submanifold of a compact holo-morphic Poisson manifold ( W, Λ ) . If H ( W, ( ∧ T W ⊕ i ∗ N W/V ) • ) = 0 , then there exists an extended Poissonanalytic family V ⊂ ( W × M , Λ) of compact holomorphic Poisson submanifolds V t , t ∈ M , of ( W, Λ t ) suchthat V = V ⊂ ( W, Λ ) and the characteristic map σ : T ( M ) → H ( W, ( ∧ T W ⊕ i ∗ N V/W ) • ) ∂∂t (cid:18) ∂ (Λ t , V t ) ∂t (cid:19) t =0 is an isomorphism.Proof. We extend the argument in the proof of Theorem 3.3.1 in the context of simultaneous deformations.We keep the notations in subsection 4.1.Let { η , ..., η ρ , ..., η l } be a basis of H ( W, ( ∧ T W ⊕ i ∗ N W/V ) • ). On each neighborhood W i (we recall U i = W i ∩ V ), η ρ is represented as η ρi = ( λ ρi ( w i , z i )) ⊕ ( γ ρi ( z i ) , ..., γ rρi ( z i )) ∈ Γ( W i , ∧ T W ) ⊕ ( ⊕ r Γ( U i , O V )) EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 35 such that λ ρi ( w i , z i ) = λ ρj ( w j , z j )(4.3.2) − [ λ ρi ( w i , z i ) , Λ ] = 0(4.3.3) γ ρi ( z ) = F ik ( z ) · γ ρk ( z ) , z ∈ U i ∩ U k (4.3.4) [ λ ρi ( w i , z i ) , w αi ] | w i =0 − [Λ , γ αρi ( z i )] | w i =0 + r X β =1 γ βρi T βiα (0 , z i ) = 0 , z i ∈ U i , α = 1 , ..., r. (4.3.5)We note that (4.3.2) implies λ ρ := { λ ρi ( w i , z i ) } ∈ H ( W, ∧ T W ).Let ǫ be a small positive number. In order to prove Theorem 4.3.1, it suffices to construct vector-valuedholomorphic functions ϕ i ( z i , t ) = ( ϕ i ( z i , t ) , ..., ϕ ri ( z i , t )) in z i and t with | z i | < , | t | < ǫ with | ϕ i ( z i , t ) | < t ) which is a convergent power series in t with coefficients in ∈ H ( W, ∧ T W ) satisfying the boundarycondition ϕ i ( z i ,
0) = 0 ,∂ϕ i ( z i , t ) ∂t ρ | t =0 = γ ρi ( z )Λ(0) = Λ ∂ Λ( t ) ∂t ρ | t =0 = λ ρ such that ϕ i ( g ik ( ϕ k ( z k , t ) , z k ) , t ) = f ik ( ϕ ( z k , t ) , z k ) , ( ϕ k ( z k , t ) , z k ) ∈ W k ∩ W i , (4.3.6) [Λ( t ) ,w αi − ϕ αi ( z i , t )] | w i = ϕ i ( z i ,t ) = 0 , α = 1 , ..., r (4.3.7) [Λ( t ) , Λ( t )] = 0(4.3.8)Recall Notation 1. Then the equalities (4 . . , (4 . . . .
8) are equivalent to the system of congruences ϕ mi ( g ik ( ϕ mk ( z k , t ) , z k ) , t ) ≡ m f ik ( ϕ mk ( z k , t ) , z k ) , m = 1 , , , · · · (4.3.9) [Λ m ( t ) ,w αi − ϕ αmi ( z i , t )] | w i = ϕ mi ( z i ,t ) ≡ m , m = 1 , , , · · · , α = 1 , ..., r, (4.3.10) [Λ m ( t ) , Λ m ( t )] ≡ m , m = 1 , , , · · · . (4.3.11)We will construct the formal power series ϕ mi ( z i , t ) and Λ m ( t ) satisfying (4 . . m , (4 . . m , and (4 . . m by induction on m .We define ϕ αi | ( z i , t ) = P lρ =1 t ρ γ αρi ( z i ), and Λ ( t ) = P lρ =1 t ρ λ ρ . Then from (4 . . ϕ i | ( z i , t ) satisfies(4 . . . On the other hand, from (4 . . ( t ) , w αi ] | w i =0 − [Λ , ϕ αi | ( z i , t )] | w i =0 + P rβ =1 ϕ βi | ( z i , t ) T βiα (0 , z i ) =0. Then we get, from (3 . . + Λ ( t ) , w αi − ϕ αi | ( z i , t )] = r X β =1 ( w βi − ϕ βi | ( z i , t )) T βiα ( w i , z i ) − [Λ ( t ) , ϕ αi | ( z i , t )] + r X β =1 w βi P βiα ( w i , z i , t )for some P βiα ( w i , z i , t ) which are homogenous polynomials of degree 1 in t , ..., t l with coefficients in Γ( U i , T W )so that we obtain [Λ ( t ) , w αi − ϕ αi | ( z i , t )] | w i = ϕ i | ( z i ,t ) ≡
0, which implies (4 . . . Lastly from (4 . . − [Λ ( t ) , Λ ] = 0 so that [Λ + Λ ( t ) , Λ + Λ ( t )] ≡ , Λ ( t )] = 0, which implies (4 . . . Hencethe induction holds for m = 1.Now we assume that we have already constructed ϕ mi ( z i , t ) = ( ϕ mi ( z i , t ) , · · · , ϕ αmi ( z i , t ) , · · · , ϕ rmi ( z i , t ))satisfying (4 . . m , (4 . . m and (4 . . m such that for α = 1 , ...r ,[Λ m ( t ) , w αi − ϕ αmi ( z i , t )] = r X β =1 ( w βi − ϕ βmi ( z i , t )) T βiα ( w i , z i ) + Q αmi ( z i , t ) + r X β =1 w βi P βmiα ( w i , z i , t )(4.3.12) such that the degree of P βmiα ( w i , z i , t ) is at least 1 in t , ..., t l . We note that we can rewrite (4.3.12) in thefollowing way. r X β =1 ( w βi − ϕ βmi ( z i , t )) T βiα ( w i , z i ) + Q αmi ( z i , t ) + r X β =1 φ βmi ( z i , t ) P βmiα ( ϕ mi ( z i , t ) , z i , t ) + r X β =1 ( w βi − ϕ βmi ( z i , t )) L βiα ( w i , z i , t )such that the degree of L βiα ( w i , z i , t ) in t , ..., t l is at least 1 so that (4.3.12) becomes the following form:[Λ m ( t ) , w αi − ϕ αmi ( z i , t )] = r X β =1 ( w βi − ϕ βmi ( z i , t )) T βiα ( w i , z i ) + K αmi ( z i , t ) + r X β =1 ( w βi − ϕ βmi ( z i , t )) L βiα ( w i , z i , t )(4.3.13)where K αmi ( z i , t ) := Q αmi ( z i , t ) + P rβ =1 φ βmi ( z i , t ) P βmiα ( ϕ mi ( z i , t ) , z i , t ).We set ψ ik ( z k , t ) := [ ϕ mi ( g ik ( ϕ mk ( z k , t ) , z k ) , t ) − f ik ( ϕ mk ( z k , t ) , z k )] m +1 (4.3.14) G αi ( z i , t ) := [[Λ m ( t ) , w αi − ϕ αmi ( z i , t )] | w i = ϕ mi ( z i ,t ) ] m +1 , α = 1 , ..., r. (4.3.15) Π( t ) := [[Λ m ( t ) , Λ m ( t )]] m +1 (4.3.16)We claim that( { (0) } , ( ψ ik ( z k , t ) , ..., ψ rik ( z k , t )) } ) ⊕ ( − Π( t ) , { ( G i ( z i , t ) , ..., G ri ( z i , t )) } ) defines a 1-cocyclein the following ˇCech resolution of ( ∧ T W ⊕ i ∗ N V/W ) • C ( U , ∧ T W ⊕ i ∗ ( N W/V ⊗ ∧ T W | V )) ˜ ∇ x C ( U , ∧ T W ⊕ i ∗ ( N W/V ⊗ T W | V )) δ −−−−→ C ( U , ∧ T W ⊕ i ∗ ( N W/V ⊗ T W | V )) ˜ ∇ x ˜ ∇ x C ( U , ∧ T W ⊕ i ∗ N W/V ) − δ −−−−→ C ( U , ∧ T W ⊕ i ∗ N W/V ) δ −−−−→ C ( U , ∧ T W ⊕ i ∗ N W/V )By defining ψ ik ( z, t ) = ψ ik ( z k , t ) for (0 , z k ) ∈ U k ∩ U i , we have the equality (see [Kod62] p.152-153) ψ ik ( z, t ) = ψ ij ( z, t ) + F ij · ψ jk ( z, t ) , for z ∈ U i ∩ U j ∩ U k (4.3.17)On the other hand, by applying [Λ m ( t ) , − ] on (4 . . m ( t ) , Λ m ( t )] , w αi − ϕ αmi ( z i , t )] = r X β =1 − [Λ m ( t ) , w βi − ϕ βmi ( z i , t )] ∧ T βiα ( w i , z i ) + r X β =1 ( w βi − ϕ βmi ( z i , t ))[Λ m ( t ) , T βiα ( w i , z i )](4.3.18)+ [Λ m ( t ) , K αmi ( z i , t )] + r X β =1 − [Λ m ( t ) , w βi − ϕ βmi ( z i , t )] ∧ L βiα ( w i , z i , t ) + r X β =1 ( w βi − ϕ βmi ( z i , t ))[Λ m ( t ) , L βiα ( w i , z i , t )]By restricting (4.3.18) to w i = ϕ mi ( z i , t ), since G αi ( z i , t ) ≡ m , α = 1 , ..., r , Π( t ) ≡ m
0, the degree L βiα ( w i , z i , t )is at least 1 in t , ..., t l and we have, from (4 . . G αi ( z i , t ) ≡ m +1 K αmi ( z i , t ) = Q αmi ( z i , t ) + r X β =1 φ βmi ( z i , t ) P βmiα ( ϕ mi ( z i , t ) , z i , t ) , (4.3.19)we obtain [ 12 Π( t ) , w αi ] | w i =0 = r X β =1 − G βi ( z i , t ) ∧ T βiα (0 , z i ) + [Λ , G αi ( z i , t )] | w i =0 (4.3.20)Next, since f αik ( w k , z k ) − ϕ αmi ( g ik ( w k , z k ) , t ) − ( f αik ( ϕ mk ( z k , t ) , z k ) − ϕ αmi ( g ik ( ϕ mk ( z k , t ) , z k ) , t ) = P rβ =1 ( w βk − ϕ βmk ( z k , t )) · S βkα ( w k , z k , t ) for some S βkα ( w k , z k , t ). By setting t = 0, we get f αik ( w k , z k ) − f αik (0 , z k ) = EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 37 P rβ =1 w βk · S βkα ( w k , z k , w γk ans setting w k = 0, we obtain ∂f αik ( w k ,z k ) ∂w γk | w i =0 = S γkα (0 , z k , m ( t ) , f αik ( w k , z k ) − ϕ αmi ( g ik ( w k , z k ) , t )] | w k = ϕ mk ( z k ,t ) + [Λ , ψ αik ( z k , t )] | w k =0 ≡ m +1 [Λ m ( t ) , f αik ( w k , z k ) − ϕ αmi ( g ik ( w k , z k ) , t )] | w k = ϕ mk ( z k ,t ) + [Λ m ( t ) , ψ αik ( z k , t )] | w k = ϕ mk ( z k ,t ) ≡ m +1 [Λ m ( t ) , f αik ( w k , z k ) − ϕ αmi ( g ik ( w k , z k ) , t ) + ψ αik ( z k , t )] | w k = ϕ mk ( z k ,t ) ≡ m +1 [Λ m ( t ) , r X β =1 ( w βk − ϕ βmk ( z k , t )) S βkα ( w k , z k , t )] | w k = ϕ mk ( z k ,t ) ≡ m +1 r X β =1 [Λ m ( t ) , w βk − ϕ βmk ( z k , t )] | w k = ϕ mk ( z k ,t ) · S βkα ( ϕ mk ( z k , t ) , z k , t ) ≡ m +1 r X β =1 G βk ( z k , t ) · S βkα (0 , z k , ≡ m +1 r X β =1 G βk ( z k , t ) · ∂f αik ( w k , z k ) ∂w βk | w k =0 Hence we obtain the equality[Λ m ( t ) , f αik ( w k , z k ) − ϕ αmi ( g ik ( w k , z k ) , t )] | w k = ϕ mk ( z k ,t ) + [Λ , ψ αik ( z k , t )] | w k =0 = r X β =1 G βk ( z k , t ) · ∂f αik ( w k , z k ) ∂w βk | w k =0 (4.3.21)On the other hand, from (4 . . m ( t ) , f αik ( w k , z k ) − ϕ αmi ( g ik ( w k , z k ) , t )] = r X β =1 ( f βik ( w k , z k ) − ϕ βmi ( g ik ( w k , z k ) , t )) T βiα ( w i , z i )(4.3.22) + K αmi ( z i , t ) + r X β =1 ( f βik ( w k , z k ) − ϕ βmi ( g ik ( w k , z k ) , t )) L βiα ( w i , z i , t )By restricting (4.3.22) to w k = ϕ mk ( z i , t ), we get[Λ m ( t ) , f αik ( w k , z k ) − ϕ αmi ( g ik ( w k , z k ) , t )] | w k = ϕ mk ( z k ,t ) ≡ m +1 r X β =1 − ψ βik ( z k , t ) T βiα (0 , z i ) + G αi ( z i , t )(4.3.23)Hence from (4 . .
21) and (4 . . r X β =1 − ψ βik ( z k , t ) T βiα (0 , z i ) + G αi ( z i , t ) + [Λ , ψ αik ( z k , t )] | w k =0 = r X β =1 G βk ( z k , t ) · ∂f αik ( w k , z k ) ∂w βk | w k =0 Lastly, [Λ , Π( t )] ≡ m +1 [Λ , [Λ m ( t ) , Λ m ( t )]] ≡ m +1 [Λ m ( t ) , [Λ m ( t ) , Λ m ( t )]] = 0 so that we get − [ −
12 Π( t ) , Λ ] = 0(4.3.24)Hence from (4 . . , (4 . . , (4 . .
21) and (4 . . { (0) } , ( ψ ik ( z k , t ) , ..., ψ rik ( z k , t )) } ) ⊕ ( −
12 Π( t ) , { ( G i ( z i , t ) , ..., G ri ( z i , t ) } ))defines a 1-cocycle in the above complex so that we get the claim. We call ψ m +1 ( t ) := { ( ψ ik ( z k , t ) , ..., ψ rik ( z k , t )) } , G m +1 ( t ) := { ( G i ( z i , t ) , ..., G ri ( z i , t )) } and Π m +1 ( t ) := − Π( t ) the m -th obstruction so that the coefficientsof (0 , ψ m +1 ( t )) ⊕ ( Π m +1 ( t ) , G m +1 ( t )) in t , ..., t l lies in H ( W, ( ∧ T W ⊕ i ∗ N V/W ) • ).On the other hand, by hypothesis, the cohomology group H ( W, ( ∧ T W ⊕ i ∗ N V/W ) • ) vanishes. There-fore there exists Λ i | m +1 ( t ) , ϕ αi | m +1 ( z i , t ) , α = 1 , ..., r such that ψ ik ( z, t ) = F ik ( z ) ϕ k | m +1 ( z, t ) − ϕ i | m +1 ( z, t ),[Λ m +1 ( t ) , w αi ] | w i =0 + P rβ =1 ϕ βi | m +1 ( z i , t ) T βiα (0 , z i ) − [Λ , ϕ αi | m +1 ( z i , t )] | w i =0 = − G αi ( z i , t ) and − [Λ m +1 ( t ) , Λ ] = Π( t ). Then we can show (4 . . m +1 (for the detail, see [Kod62] p.154). On the other hand,[Λ m +1 ( t ) , w αi ] − [Λ , ϕ αi | m +1 ( z i , t )] = − r X β =1 ϕ βi | m +1 ( z i , t ) T βiα ( w i , z i ) − G αi ( z i , t ) + r X β =1 w βi R βiα ( w i , z i , t )(4.3.25)where the degree of R βiα ( w i , z i , t ) is m + 1 in t . Let [Λ m ( t ) − Λ , − ϕ i | m +1 ( z i , t )] + [Λ | m +1 ( t ) , − ϕ αmi ( z i , t ) − ϕ i | m +1 ( z i , t )] = H αmi ( z i , t ) + P rβ =1 w βi M βiα ( w i , z i , t ), where the degree of H αmi ( z i , t ) and M βiα ( w i , z i , t ) is atleast m + 2. Then from (4.3.12) and (4.3.25), we have[Λ m ( t ) + Λ m +1 ( t ) , w αi − ϕ αmi ( z i , t ) − ϕ αi | m +1 ( z i , t )](4.3.26) = [Λ m ( t ) , w αi − ϕ αmi ( z i , t )] + [Λ m ( t ) − Λ , − ϕ αi | m +1 ( z i , t )] + [Λ , − ϕ αi | m +1 ( z i , t )]+ [Λ m +1 ( t ) , w αi ] + [Λ m +1 ( t ) , − ϕ αmi ( z i , t ) − ϕ i | m +1 ( z i , t )]= r X β =1 ( w βi − ϕ βmi − ϕ βi | m +1 ) T βiα ( w i , z i ) + Q αmi ( z i , t ) − G αi ( z i , t ) + H αmi ( z i , t )+ r X β =1 w αi ( P βmiα ( w i , z i , t ) + R βiα ( w i , z i , t ) + M βiα ( w i , z i , t ))We show that [Λ m +1 ( t ) , w αi − ϕ αm +1 i ] | w i = ϕ αm +1 i ≡ m +1
0. Indeed, by restricting (4 . .
26) to w i = ϕ m +1 i ( z i , t ),we get, from (4 . . m +1 ( t ) , w αi − ϕ α ( m +1) i ( z i , t )] | w i = ϕ m +1 i ( z i ,t ) ≡ m +1 Q αmi ( z i , t ) − G αi ( z i , t ) + r X β =1 ( ϕ αmi ( z i , t ) + ϕ αi | m +1 ( z i , t )) P βmiα ( ϕ mi ( z i , t ) + ϕ i | m +1 ( z i , t ) , z i , t ) ≡ m +1 Q αmi ( z i , t ) − G αi ( z i , t ) + r X β =1 ϕ αmi ( z i , t ) P βmiα ( ϕ mi ( z i , t ) , z i , t ) ≡ m +1 K αmi ( z i , t ) − G αi ( z i , t ) ≡ m +1 . . m +1 . Lastly [Λ m ( t )+Λ m +1 ( t ) , Λ m ( t )+Λ m +1 ( t )] ≡ m +1 [Λ m ( t ) , Λ m ( t )]+2[Λ , Λ m +1 ( t )] ≡ m +1 [Λ m ( t ) , Λ m ( t )] − Π( t ) ≡ m +1 . . m +1 . This completes the inductive constructions of ϕ mi ( z i , t ) , i ∈ I , and Λ m ( t ).4.4. Proof of convergence.
We will show that we can choose ϕ i | m ( z i , t ) and Λ m ( t ) in each inductive step so that the formal powerseries ϕ i ( z i , t ) , i ∈ I and Λ( t ) constructed in the previous subsection, converges absolutely for | t | < ǫ for asufficiently small number ǫ > = Λ i ( w i , z i ) = P r + dp,q =1 Λ ipq ( w i , z i ) ∂∂x pi ∧ ∂∂x qi with Λ ipq ( w i , z i ) = − Λ ipq ( w i , z i ), where x i = ( w i , z i ) on W i , and W δi is the subdomain of W i consisting of allpoints ( w i , z i ), | w i | < − δ, | z i | < − δ for a sufficiently small number δ > { W δi | i ∈ I } forms acovering of W , and { U δi = W δi ∩ V | i ∈ I } forms a covering of V . Recall Notation 3. We denote Λ m ( t ) on W i by Λ mi ( w i , z i , t ) and Π m ( t ) on W i by Π mi ( w i , z i , t ). Then Λ mi ( w i , z i , t ) is of the formΛ mi ( w i , z i , t ) = r + d X p,q =1 Λ impq ( w i , z i , t ) ∂∂x pi ∧ ∂∂x qi , Λ impq ( w i , z i , t ) = − Λ imqp ( w i , z i , t )(4.4.1)such that Λ i pq ( w i , z i , t ) = Λ ipq ( w i , z i ). We may assume that | F λikµ (0 , z ) | < c with c >
1. Then ϕ i | ( z i , t ) ≪ A ( t ) and Λ i | ( w i , z i , t ) ≪ A ( t ) if b is sufficiently large. Now, assuming the inequalities ϕ mi ( z i , t ) ≪ A ( t ) , Λ mi ( w i , z i , t ) − Λ i ( w i , z i ) ≪ A ( t ) , ( w i , z i ) ∈ W δi (4.4.2)for an integer m ≥
1, we will estimate the coefficients of the homogenous polynomials ψ ik ( z, t ) , Π i ( w i , z i , t ),and G αi ( z i , t ) from (4 . . , (4 . . . . ψ ik ( z, t ). As in (3.4.4), we have ψ ik ( z k , t ) ≪ c A ( t ) , z k ∈ U k ∩ U i , (4.4.3) EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 39 where c = 2 rc c rab (cid:16) d δ + rc (cid:17) b > max { c ra, c raδ } . (4.4.4)Next we estimate G αi ( z i , t ). We note that G αi ( z i , t ) ≡ m +1 [Λ mi ( w i , z i , t ) , w αi − ϕ αmi ( z i , t )] | w i = ϕ mi ( z i ,t ) (4.4.5) ≡ m +1 [Λ mi ( w i , z i , t ) − Λ i ( w i , z i ) , w αi ] | w i = ϕ mi ( z i ,t ) − [Λ mi ( w i , z i , t ) − Λ i ( w i , z i ) , ϕ αmi ( z i , t )] | w i = ϕ mi ( z i ,t ) + [Λ i ( w i , z i ) , w αi − ϕ αmi ( z i , t )] | w i = ϕ mi ( z i ,t ) We estimate each term in (4 . . mi ( w i , z i , t ) − Λ i ( w i , z i ) , w αi ] | w i = ϕ mi ( z i ,t ) ] m +1 in(4 . . d + r X p,q =1 (Λ impq ( w i , z i , t ) − Λ ipq ( w i , z i )) ∂∂x pi ∧ ∂∂x qi , w αi ] | w i = ϕ mi ( z i ,t ) = d + r X p,q =1 impq ( ϕ mi ( z i , t ) , z i , t ) − Λ ipq ( ϕ mi ( z i , t ) , z i )) ∂w αi ∂x pi ∂∂x qi (4.4.6)On the other hand, Φ impq ( w i , z i , t ) := Λ impq ( w i , z i , t ) − Λ ipq ( w i , z i ) ≪ A ( t ) from (4 . .
2) and has the degree ≤ m in t . For any p, q , we expand Φ impq ( w i , z i , t ) into power series in w i , ...w ri whose coefficients are holomorphicfunctions of z = (0 , z i ) defined on U i :Φ impq ( w i , z i , t ) = X µ ,...,µ r ≥ Φ impq,µ ,...,µ r ( z i , t ) w µ i · · · w rµ r i (4.4.7)If ( w i , z i ) ∈ W δi , we have, by Cauchy’s integral formula,Φ impq,µ ,...,µ r ( z i , t ) = (cid:18) πi (cid:19) r Z | ξ − w i | = δ · · · Z | ξ r − w ri | = δ Φ impq ( w i , z i , t )( ξ − w i ) µ +1 · · · ( ξ r − w ri ) µ r +1 dξ · · · dξ r so that we get Φ impq,µ ,...,µ r ( z i , t ) ≪ A ( t ) 1 δ µ + ··· + µ r (4.4.8)Since constant terms of Φ impq ( w i , z i , t ) with respect to w i , ..., w ri does not contribute to [[Λ mi ( w i , z i , t ) − Λ i ( w i , z i ) , w αi ] | w i = ϕ mi ( z i ,t ) ] m +1 , from (4 . .
7) and (4 . . abδ < , (for the detail, see[Kod05] p.300) Φ impq ( ϕ mi ( z i , t ) , z i , t ) ≪ A ( t ) X µ + ... + µ r ≥ (cid:18) A ( t ) δ (cid:19) µ + ··· + µ r ≪ r +1 δ A ( t ) ≪ r +1 abδ A ( t )(4.4.9)Then from (4 . .
6) and (4 . . mi ( w i , z i , t ) − Λ i ( w i , z i ) , w αi ] | w i = ϕ mi ( z i ,t ) ] m +1 ≪ d + r ) r +1 abδ A ( t )(4.4.10)Next we estimate [[Λ mi ( w i , z i , t ) − Λ i ( w i , z i ) , ϕ αmi ( z i , t )] | w i = ϕ mi ( z i ,t ) ] m +1 in (4 . . . .
2) and(3 . . d + r X p,q =1 (Λ impq ( w i , z i , t ) − Λ ipq ( w i , z i )) ∂∂x pi ∧ ∂∂x qi , ϕ αmi ( z i , t )] | w i = ϕ mi ( z i ,t ) (4.4.11) = d + r X p,q =1 impq ( ϕ mi ( z i , t ) , z i , t ) − Λ ipq ( ϕ mi ( z i , t ) , z i )) ∂ϕ αmi ( z i , t ) ∂x pi ∂∂x qi ≪ d + r ) A ( t ) A ( t ) δ ≪ d + r ) abδ A ( t )Hence we get [[Λ mi ( w i , z i , t ) − Λ i ( w i , z i ) , ϕ αmi ( z i , t )] | w i = ϕ mi ( z i ,t ) ] m +1 ≪ d + r ) abδ A ( t )(4.4.12) We estimate [Λ i ( w i , z i ) , w αi − ϕ αmi ( z i , t )] | w i = ϕ mi ( z i ,t ) in (4 . . . . i ( w i , z i ) , w αi − ϕ αmi ( z i , t )] | w i = ϕ mi ( z i ,t ) ] m +1 ≪ e A ( t ) , z ∈ U δi (4.4.13)where e = d + r ) e r ab + r + d ) e raδb with b > e ra .Hence from (4 . . , (4 . . , (4 . . G αi ( z i , t ) = [[Λ mi ( w i , z i , t ) , w αi − ϕ αmi ( z i , t )] | w i = ϕ mi ( z i ,t ) ] m +1 ≪ e A ( t )(4.4.14)where e = ( d + r ) r +2 abδ + d + r ) abδ + e with b > max { aδ , e ra } (4.4.15)Next we estimate Π i ( w i , z i , t ) = [[Λ mi ( w i , z i , t ) , Λ mi ( w i , z i , t )]] m +1 . Since Π i ( w i , z i , t ) ≡ m i ( w i , z i , t ) ≡ m +1 [Λ mi ( w i , z i , t ) , Λ mi ( w i , z i , t )] ≡ m +1 [Λ mi ( w i , z i , t ) − Λ i ( w i , z i ) , Λ mi ( w i , z i , t ) − Λ i ( w i , z i )] + 2[Λ i ( w i , z i ) , Λ mi ( w i , z i ) − Λ i ( w i , z i )] , we have Π i ( w i , z i , t ) = [[Λ mi ( w i , z i , t ) − Λ i ( w i , z i ) , Λ mi ( w i , z i , t ) − Λ i ( w i , z i )]] m +1 (4.4.16)We note the following two remarks. Remark 4.4.17.
Let σ = P p,q σ pq ∂∂x p ∧ ∂∂x q with σ pq = − σ qp and φ = P l,k φ lk ∂∂x l ∧ ∂∂x k with φ lk = − φ kl .Then [ σ, φ ] = P p,q,l,k [ σ pq ∂∂x p ∧ ∂∂x q , φ lk ∂∂x l ∧ ∂∂x k ] = P p,q,l,k [ σ pq ∂∂x p , φ lk ∂∂x l ] ∂∂x q ∧ ∂∂x k − φ lk [ σ pq ∂∂x p , ∂∂x k ] ∂∂x q ∧ ∂∂x l − σ pq [ ∂∂x q , φ lk ∂∂x l ] ∂∂x p ∧ ∂∂x k = P p,q,l,k σ pq ∂φ lk ∂x p ∂∂x l ∧ ∂∂x q ∧ ∂∂x k − φ lk ∂σ pq ∂x l ∂∂x p ∧ ∂∂x q ∧ ∂∂x k + φ lk ∂σ pq ∂x k ∂∂x p ∧ ∂∂x q ∧ ∂∂x l − σ pq ∂φ lk ∂x q ∂∂x l ∧ ∂∂x p ∧ ∂∂x k Remark 4.4.18. ∂ (Λ impq ( x i , t ) − Λ ipq ( x i )) ∂x si = 12 πi Z | ξ − x si | = δ Λ impq ( x i , ... s − th ξ , ..., x d + ri , t ) − Λ ipq ( x i , ..., s − th ξ , ..., x d + ri )( ξ − x si ) dξ ≪ A ( t ) δ , x i ∈ W δi By Remark 4.4.17, Remark 4.4.18 and (4 . . mi ( w i , z i , t ) − Λ i ( w i , z i ) , Λ mi ( w i , z i , t ) − Λ i ( w i , z i )](4.4.19) = [ d + r X p,q =1 (Λ impq ( w i , z i , t ) − Λ ipq ( w i , z i )) ∂∂x pi ∧ ∂∂x qi , d + r X p,q =1 (Λ impq ( w i , z i , t ) − Λ ipq ( w i , z i )) ∂∂x pi ∧ ∂∂x qi ] ≪ d + r ) A ( t ) · A ( t ) δ ≪ d + r ) aδb A ( t )Hence from (4 . .
16) and (4 . . i ( w i , z i , t ) ≪ d + r ) A ( t ) · A ( t ) δ ≪ d + r ) aδb A ( t ) = e A ( t )(4.4.20)where e = d + r ) aδb . Notation 4.
We consider P k = P d + rα,β =1 P kαβ ( x k ) ∂∂x αk ∧ ∂∂x βk ∈ Γ( W i , ∧ T W ) to be a vector-valued holomorphicfunction P k ( x ) = ( P kαβ ( x k )) α,β =1 ,...,d + r on W k . On W i ∩ W k , P k is translated to P d + rα,β,p,q =1 P kαβ ( x k ) ∂h pik ∂x αk ∂h qik ∂x βk ∂∂x pi ∧ ∂∂x qi which corresponds to a vector-valued holomorphic function ( P d + rα,β =1 P kαβ ( x k ) ∂h pik ∂x αk ∂h qik ∂x βk ) p,q =1 ,...,d + r on W i denoted by H ik ( x ) P k ( x ) . EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 41
Lemma 4.4.21.
We can choose the homogenous polynomials ϕ i | m +1 ( z, t ) , Λ i | m +1 ( w, z, t ) , i ∈ I , satisfying ψ ik ( z, t ) = F ik ( z ) ϕ k | m +1 ( z, t ) − ϕ i | m +1 ( z, t ) − G αi ( z, t ) = [Λ i | m +1 ( w, z, t ) , w αi ] | w i =0 − [ ϕ αi | m +1 ( z, t ) , Λ ] | w i =0 + r X β =1 ϕ i | m +1 ( z, t ) T βiα (0 , z )12 Π i ( w i , z i , t ) = − [Λ i | m +1 ( w, z, t ) , Λ ](Θ ik ( w, z, t ) = H ik ( w, z )Λ k | m +1 ( w, z, t ) − Λ i | m +1 ( w, z, t ) = 0) in such a way that ϕ i | m +1 ( z, t ) ≪ c ( c + e + e ) A ( t ) and Λ i | m +1 ( w, z, t ) ≪ c ( c + e + e ) A ( t ) for ( w, z ) ∈ W δi , where c is independent of m .Proof. For any 0-cochain ( π, ϕ ) = ( { π i } , { ϕ i } ), and 1-cochain (Θ , ψ ) ⊕ (Π , G ) = ( { Θ ik ( w, z ) } , { ψ ik ( z ) } ) ⊕ ( { Π i ( w, z ) } , { P rα =1 G αi ( z ) e αi } ), we define the norms of ( π, ϕ ) and (Θ , ψ ) ⊕ (Π , G ) by || ( π, ϕ ) || := max i sup x =( w,z ) ∈ W i | π i ( w, z ) | + max i sup z ∈ U i | ϕ i ( z ) | , || (Θ , ψ ) ⊕ (Π , G ) || := max i,k sup x =( w,z ) ∈ W i ∩ W k | Θ ik ( w, z ) | + max i sup ( w,z ) ∈ W δi | Π i ( w, z ) | + max i,k sup z ∈ U i ∩ U k | ψ ik ( z ) | + max i,α sup z ∈ U δi | G αi ( z ) | The coboundary of ( π, ϕ ) is defined by˜ δ ( π, ϕ ) := ( {− H ik ( x ) π k ( x ) + π i ( x ) } , {− F ik ( z ) ϕ k ( z ) + ϕ i ( z ) } ) ⊕ ( {− [ π i ( x ) , Λ ] } , {⊕ rα =1 [ π i ( x ) , w αi ] | w i =0 − [ ϕ αi ( z ) , Λ ] | w i =0 + r X β =1 ϕ βi ( z ) T βiα (0 , z ) e αi } )For any coboundary of the form (0 , ψ ) ⊕ (Π , G ), we define ι ((0 , ψ ) ⊕ (Π , G )) = inf ˜ δ ( π,ϕ )=(0 ,ψ ) ⊕ (Π ,G ) || ( π, ϕ ) || To prove Lemma 4.4.21, it suffices to show the existence of a constant c such that ι ((0 , ψ ) ⊕ (Π , G )) ≤ c || (0 , ψ ) ⊕ (Π , G ) || . Assume that such a constant does not exist. Then we can find a sequence (0 , ψ ( µ ) ) ⊕ (Π ( µ ) , G ( µ ) ) such that ι ((0 , ψ ( µ ) ) ⊕ (Π ( µ ) , G ( µ ) )) = 1 , || (0 , ψ ( µ ) ) ⊕ (Π ( µ ) , G ( µ ) ) || < µ Then there exists ( π ( µ ) , ϕ ( µ ) ) with δ ( π ( µ ) , ϕ ( µ ) ) = (0 , ψ ( µ ) ) ⊕ (Π ( µ ) , G ( µ ) ) satisfying || π ( µ ) || < || ϕ ( µ ) || <
2. We note that π ( µ ) is a global bivector field in H ( W, ∧ T W ). We take a covering { ¯ W δi } of W and acovering { ¯ U δi = ¯ W δi ∩ U i } of V . Since | π µk ( x ) | < x ∈ W k and | φ µk ( z ) | < z ∈ U k = W k ∩ V ,there exists a subsequence ( π ( µ ) , ϕ ( µ ) ) , ( π ( µ ) , ϕ ( µ ) ) , · · · , ( π ( µ v ) , ϕ ( µ v ) ) , · · · of ( π ( µ ) , ϕ ( µ ) ) such that π ( µ v ) k converges absolutely and uniformly on ¯ W δk for each k , and ϕ ( µ v ) k converges absolutely and uniformly on¯ U δk . Since W is compact, we can choose a subsequence that works for all k . On the other hand, since || (0 , ψ ( µ ) ) ⊕ (Π ( µ ) , G ( µ ) ) || < µ , we have H ik ( x ) π ( µ ) k ( x ) = π ( µ ) i ( x ) , x ∈ W i ∩ W k , | F ik ( z ) ϕ ( µ ) k ( z ) − ϕ ( µ ) i ( z ) | < µ , z ∈ U i ∩ U k (4.4.22) | − [ π ( µ ) i ( x ) , Λ ] | < µ , x ∈ W δi , | [ π ( µ v ) i , w αi ] | w i =0 − [ ϕ α ( µ v ) i , Λ ] | w i =0 + r X β =1 ϕ β ( µ ) i T βiα (0 , z ) | < µ , z ∈ U δi Let π i ( x ) = lim v π ( µ v ) i ( x ) and ϕ k ( z ) = lim v ϕ ( µ v ) i ( z ). Since π ( µ v ) i converges absolutely and uniformly on W δi , π i is holomorphic on W δi . Since { W δi } covers W , and π i ( x ) = H ik ( x ) π k ( x ) for x ∈ W i ∩ W k , weget { π i ( x ) } ∈ H ( W, ∧ T W ). On the other hand, ϕ µ v i ( z ) converges absolutely and uniformly on U i . Let π := { π i ( x ) } and ϕ := { ϕ i ( z ) } . Then we have || ( π ( µ v ) − π, ϕ ( µ v ) − ϕ ) || → n → ∞ . On the other hand,by (4 . . δ ( π, ϕ ) = (0 , ⊕ ( − [ π, Λ ] , { G i,π,ϕ } ), where − [ π, Λ ]( x ) = 0 for x ∈ W δi (hence − [ π, Λ ] = 0)and G i,π,ϕ ( z ) = 0 for z ∈ U δi (hence G i,π,ϕ = 0 by identity theorem) so that ˜ δ ( π, ϕ ) = (0 , ⊕ (0 , Hence we have ˜ δ ( π ( µ v ) , ϕ ( µ v ) ) = ˜ δ ( π ( µ v ) − π, ϕ ( µ v ) − φ ) = (0 , ψ ( µ v ) ) ⊕ (Π ( µ v ) , G ( µ v ) ) which contradicts to ι ((0 , ψ ( µ v ) ) ⊕ (Π ( µ v ) , G ( µ v ) )) = 1. (cid:3) From (4 . . , (4 . .
14) and (4 . . c ( c + e + e ) = c c + c e + c e (4.4.23)= 8 c c c r ab (cid:18) d δ + rc (cid:19) + c (cid:18) ( d + r ) r +2 abδ + 2( d + r ) abδ (cid:19) + c (cid:18) d + r ) e r ab + 4( r + d ) e raδb (cid:19) + c d + r ) aδb . From (4 . . . . . .
23) and Lemma 4.4.21, by assuming b > c c c r a (cid:18) d δ + rc (cid:19) + c (cid:18) ( d + r ) r +2 aδ + 2( d + r ) aδ (cid:19) + c (cid:18) d + r ) e r a + 4( r + d ) e raδ (cid:19) + c d + r ) aδ + max { c ra, c raδ , aδ , e ra } , we can choose ϕ i | m +1 ( z i , t ) ≪ A ( t ) and Λ i | m +1 ( w i , z i , t ) ≪ A ( t ) and so ϕ i ( z i , t ) ≪ A ( t ), and Λ i ( w i , z i , t ) − Λ i ( w i , z i ) ≪ A ( t ) so that ϕ i ( z i , t ) and Λ i ( w i , z i , t ) converges for | t | < lb . Then we obtain the equality ϕ i ( g ik ( ϕ k ( z k , t ) , z k ) , t ) = f ik ( ϕ k ( z k , t ) , z k ) , for | t | < ǫ, ( ϕ k ( z k , t ) , z k ) ∈ W δi ∩ W δk [Λ i ( w i , z i , t ) , w αi − ϕ αi ( z i , t )] | w i = ϕ i ( z i ,t ) = 0[Λ i ( w i , z i , t ) , Λ i ( w i , z i , t )] = 0for a sufficiently small number ǫ >
0. This completes the proof of Theorem 4.3.1. (cid:3)
In the case H ( W, ( ∧ T W ⊕ i ∗ N V/W ) • ) = 0, our proof of Theorem 4.3.1 also proves the following: Theorem 4.4.24.
If the obstruction (0 , ψ m +1 ( t )) ⊕ ( Π m +1 ( t ) , G m +1 ( t )) vanishes for each integer m ≥ ,then there exists an extended Poisson analytic family V of compact holomorphic Poisson submanifolds V t , t ∈ M , of ( W, Λ t ) such that V = V ⊂ ( W, Λ ) and the characteristic map σ : T ( M ) → H ( W, ( ∧ T W ⊕ i ∗ N V/W ) • ) ∂∂t (cid:18) ∂ (Λ t , V t ) ∂t (cid:19) t =0 is an isomorphism. Maximal families: Theorem of completeness.Definition 4.5.1.
Let
V ⊂ ( W × M, Λ) ω −→ M be an extended Poisson analytic family of compact holomorphicPoisson submanifolds of W so that ω − ( t ) = V t is a compact holomorphic Poisson submanifold of ( W, Λ t ) , t ∈ M and let t be a point on M . We say that V ω −→ M is maximal at t if, for any extended Poisson analyticfamily V ′ ⊂ ( W × M ′ , Λ ′ ) ω ′ −→ M ′ of compact holomorphic Poisson submanifolds of W such that Λ t = Λ ′ t ′ and ω − ( t ) = ω ′− ( t ′ ) , t ′ ∈ M ′ , there exists a holomorphic map h of a neighborhood N ′ of t ′ on M ′ into M which maps t ′ to t such that ω ′− ( t ′ ) = ω − ( h ( t ′ )) and Λ ′ t ′ = Λ h ( t ′ ) for t ′ ∈ N ′ . We note that if weset a holomorphic map ˆ h : W × N ′ → W × M defined by ( w, t ′ ) → ( w, h ( t ′ )) , then ˆ h is a Poisson map ( W × N ′ , Λ ′ ) → ( W × M, Λ) and the restriction map of ˆ h to V ′ | N ′ = ω ′− ( N ′ ) ⊂ ( W × N ′ , Λ ′ ) defines aPoisson map V ′ | N ′ → V so that V ′ | N ′ is the family induced from V by h , which means V ω −→ M is completeat t . Theorem 4.5.2 (theorem of completeness) . Let
V ⊂ ( W × M , Λ) be an extended Poisson analytic familyof compact holomorphic Poisson submanifolds V t of ( W, Λ t ) . If the characteristic map ρ : T ( M ) → H ( W, ( ∧ T W ⊕ i ∗ N W/V ) • ) ∂∂t (cid:18) ∂ (Λ t , V t ) ∂t (cid:19) t =0 is bijective, then the family V is maximal at t = 0 . EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 43
Proof.
Consider an arbitrary extended Poisson analytic family V ′ ⊂ ( W × M, Λ ′ ) of compact holomorphicPoisson submanifolds V ′ s , s ∈ M ′ of ( W, Λ ′ s ), where M ′ = { s = ( s , ..., s q ) ∈ C q || s | < } . We will construct aholomorphic map h : s → t = h ( s ) of a neighborhood N ′ of 0 in M ′ into M with h (0) = 0, V ′ s = V h ( s ) andΛ ′ s = Λ h ( s ) .We keep the notations in subsection 4.2 so that the holomorphic Poisson submanifold V t of ( W, Λ t ) isdefined on each domain W i , i ∈ I by the equation w i = ϕ i ( z i , t ) and satisfy[Λ i ( w i , z i , t ) , w αi − ϕ αi ( z i , t )] = r X β =1 ( w βi − ϕ βi ( z i , t )) T βiα ( w i , z i , t )(4.5.3)We may assume that V ′ s is defined in each domain W i , i ∈ I by w i = θ i ( z i , s ) where θ i ( z i , s ) is a vector-valuedholomorphic function of z i and s, | z i | < , | s | <
1, and let Λ ′ i ( w i , z i , s ) be the Poisson structure on W i × M ′ induced from Λ ′ . Then we have[Λ ′ i ( w i , z i , s ) , w αi − θ αi ( z i , s )] = r X β =1 ( w βi − θ βi ( z i , s )) P βiα ( w i , z i , s )(4.5.4)for some P βiα ( w i , z i , s ) which are power series in s with coefficients in Γ( W i , T W ) and P βiα (0 , z i ,
0) = T βiα (0 , z i ).Then V ′ s = V h ( s ) and Λ ′ s = Λ h ( s ) are equivalent to the simultaneous equations θ i ( z i , s ) = ϕ i ( z i , h ( s )) , Λ ′ i ( w i , z i , s ) = Λ i ( w i , z i , h ( s )) , i ∈ I, m = 1 , , , ... (4.5.5)Recall Notation 1 and let us write h ( s ) = h ( s )+ h ( s )+ · · · , ϕ i ( z i , t ) = ϕ i | ( z i , t )+ ϕ i | ( z i , t )+ · · · , θ i ( z i , s ) = θ i | ( z i , s ) + θ i | ( z i , s ) + · · · , Λ i ( w i , z i , t ) = Λ i ( w i , z i ) + Λ i | ( w i , z i , t ) + Λ i | ( w i , z i , t ) + · · · , and Λ ′ i ( w i , z i , s ) =Λ i ( w i , z i )+ Λ ′ i | ( w i , z i , s )+ Λ ′ i | ( w i , z i , s )+ · · · . We will construct h ( s ) satisfying (4 . .
5) by solving the systemof congruences by induction on mθ i ( z i , s ) ≡ m ϕ i ( z i , h m ( s )) , Λ ′ i ( w i , z i , s ) ≡ m Λ i ( w i , z i , h m ( s )) , i ∈ I, m = 1 , , , · · · (4.5.6)Since σ : T ( M ) → H ( W, ( ∧ T W ⊕ i ∗ N V /W ) • ) is an isomorphism by the hypothesis, any element( { B i ( w i , z i ) } , { ω i ( z i ) } ) ∈ H ( W, ( ∧ T W ⊕ i ∗ N V /W ) • ) can be written uniquely in the form ω i ( z ) = ϕ i | ( z i , u ) = l X α =1 ∂ϕ i ( z i , t ) ∂t α | t =0 u α , B i ( w i , z i ) = Λ i | ( w i , z i , u ) = l X α =1 ∂ Λ i ( w i , z i , t ) ∂t α | t =0 u α for some constant u = ( u , ..., u l ). Hence since ( { Λ ′ i | ( w i , z i , s ) } , { θ i | ( z i , s ) } ) , i ∈ I represents a linearform in s whose coefficients are in H ( W, ( ∧ T W ⊕ i ∗ N V /W ) • ), there exists a linear vector-valued func-tion h ( s ) of s such that θ i | ( z i , s ) = ϕ i | ( z i , h ( s )), and Λ ′ i | ( w i , z i , s ) = Λ i | ( w i , z i , h ( s )). This shows(4 . . . Now suppose that we have already constructed h m ( s ) satisfying (4 . . m . We will find h m +1 ( s )such that h m +1 ( s ) = h m ( s ) + h m +1 ( s ) satisfy (4 . . m +1 . Let ω i ( z i , s ) = [ θ i ( z i , s ) − ϕ i ( z i , h m ( s ))] m +1 , and B i ( w i , z i , s ) = [Λ ′ i ( w i , z i , s ) − Λ i ( w i , z i , h m ( s ))] m +1 . We claim that ω i ( z i , s ) = F ik ( z ) · ω k ( z k , s )(4.5.7) [ B i ( w i , z i , s ) , w αi ] | w i =0 − [ ω αi ( z i , s ) , Λ ] | w i =0 + r X β =1 ω βi ( z i , s ) T βiα ( z i ) = 0(4.5.8) B i ( w i , z i , s ) − B j ( w j , z j , s ) = 0(4.5.9) − [ B i ( w i , z i , s ) , Λ ] = 0(4.5.10)(4.5.7) follows from [Kod62] p.160. Since Λ ′ i ( w i , z i , s ) = Λ ′ j ( w j , z j , s ) and Λ i ( w i , z i , t ) = Λ j ( w j , z j , t ), we get(4 . . , B i ( w i , z i , s )] ≡ m +1 [Λ ′ i ( w i , z i , s ) + Λ i ( w i , z i , h m ( s )) , Λ ′ i ( w i , z i , s ) − Λ i ( w i , z i , h m ( s ))] = 0, we get (4 . . . .
3) and (4.5.4), we have[Λ , ω αi ( z i , s )] | w i =0 ≡ m +1 [Λ ′ i ( w i , z i , s ) , ω αi ( z i , s )] | w i = θ i ( z i ,s ) ≡ m +1 [Λ ′ i ( w i , z i , s ) , θ αi ( z i , s ) − w αi + w αi − ϕ αi ( z i , h m ( s ))] | w i = θ i ( z i ,s ) ≡ m +1 − [Λ ′ i ( w i , z i , s ) , w αi − θ αi ( z i , s )] | w i = θ i ( z i ,s ) + [Λ ′ i ( w i , z i , s ) , w αi − ϕ αi ( z i , h m ( s ))] | w i = θ i ( z i ,s ) ≡ m +1 [ B i ( w i , z i , s ) , w αi − ϕ αi ( z i , h m ( s ))] | w i = θ i ( z i ,s ) + [Λ i ( w i , z i , h m ( s )) , w αi − ϕ αi ( z i , h m ( s ))] | w i = θ i ( z i ,s ) ≡ m +1 [ B i ( w i , z i , s ) , w αi ] | w i =0 + r X β =1 ( θ βi ( z i , s ) − ϕ βi ( z i , h ( s ))) T βiα ( θ i ( z i , s ) , z i , h ( s )) ≡ m +1 [ B i ( w i , z i , s ) , w αi ] | w i =0 + r X β =1 ω βi ( z i , s ) T βiα (0 , z i )This proves (4 . . . . , (4 . . , (4 . . . . { B i ( w i , z i , s ) } , { ω i ( z i , s ) } ) is a homogenouspolynomial of degree m +1 in s with coefficients in H ( W, ( ∧ T W ⊕ i ∗ N V /W ) • ) so that there exists a homoge-nous polynomial h m +1 ( s ) of degree m + 1 in s such that ω i ( z i , s ) = ϕ i | ( z i , h m +1 ( s )), and B i ( w i , z i , s ) =Λ i | ( w i , z i , h m +1 ( s )) so that we have ϕ i ( z i , h m +1 ( s )) ≡ m +1 ϕ i ( z i , h m ( s )) + ω i ( z i , s ) ≡ m +1 θ i ( z i , s ), andΛ i ( w i , z i , h m +1 ( s )) ≡ m +1 Λ i ( w i , z i , h m ( s )) + B i ( w i , z i , s ) ≡ m +1 Λ ′ i ( w i , z i , s ), which completes the inductiveconstruction of h m +1 ( s ) satisfying (4 . . m +1 .4.6. Proof of convergence.
The convergence of the power series h ( s ) follows from the same arguments in [Kod62]. This completesthe proof of Theorem 4.5.2. (cid:3) Example 9.
We describe holomorphic Poisson structures on rational ruled surfaces F m = P ( O P C ( m ) ⊕O P C ) , m ≥ explictly. F m can be represented in the following way. Take two copies of U i × P C , i = 1 , ,where U i = C and write the coordinates as ( z, [ ξ , ξ ]) and ( z ′ , [ ξ ′ , ξ ′ ]) . Patch U i × P C , i = 1 , by the relation z ′ = z and [ ξ ′ , ξ ′ ] = [ ξ , z m ξ ] . We set ξ = ξ ξ and ξ ′ = ξ ′ ξ ′ . Then we have ∂∂z ′ = − z ∂∂z + mzξ ∂∂ξ and ∂∂ξ ′ = z − m ∂∂ξ so that ∂∂z ′ ∧ ∂∂ξ ′ = − z − m +2 ∂∂z ∧ ∂∂ξ . We note that a holomorphic bivector field on U × P is of the form ( d ( z ) + e ( z ) ξ + f ( z ) ξ ) ∂∂z ∧ ∂∂ξ , and a holomorphic bivector field on U × P C is of the form ( p ( z ′ ) + q ( z ′ ) ξ ′ + r ( z ′ ) ξ ′ ) ∂∂z ′ ∧ ∂∂ξ ′ , where d ( z ) , e ( z ) , f ( z ) are entire functions of z and p ( z ′ ) , q ( z ′ ) , r ( z ′ ) areentire functions of z ′ . For a holomorphic bivector field on F m which has the form on each U i × P C , we musthave d ( z ) + e ( z ) ξ + f ( z ) ξ = − ( p ( z ) + q ( z ) z m ξ + r ( z ) z m ξ ) z − m +2 = − p ( z ) z − m +2 − q ( z ) z ξ − r ( z ) z m +2 ξ so that d ( z ) = − p (cid:18) z (cid:19) z − m +2 , e ( z ) = − q (cid:18) z (cid:19) z , f ( z ) = − r (cid:18) z (cid:19) z m +2 (1) In the case of m = 0 , we have d ( z ) = a + a z + a z , e ( z ) = b + b z + b z , f ( z ) = c + c z + c z so that H ( F , ∧ T F ) ∼ = C . (2) In the case of m = 1 , we have d ( z ) = a + a z, e ( z ) = b + b z + b z , f ( z ) = c + c z + c z + c z so that H ( F , ∧ T F ) ∼ = C . (3) In the case of m = 2 , we have d ( z ) = a , e ( z ) = b + b z + b z , f ( z ) = c + c z + c z + c z + c z so that H ( F , ∧ T F ) ∼ = C . (4) In the case of m ≥ , we have d ( z ) = 0 , e ( z ) = b + b z + b z , f ( z ) = c + c z + · · · c m +2 z m +2 sothat H ( F m , ∧ T F m ) ∼ = C m +6 . In the sequel, we keep the notations in Example 9.
Example 10.
Let us consider a rational ruled surface F ∼ = P C × P C . Let us consider the Poisson structure Λ = ξ ∂∂z ∧ ∂∂ξ . Then ξ = 0 defines a holomorphic Poisson submanifold on F which is a nonsingular rationalcurve ∼ = P C and the normal bundle is N P C /F ∼ = O P C . We compute H ( F , ( ∧ T F ⊕ i ∗ N P C /F ) • ) which is the EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 45 kernel of ˜ ∇ : H ( F , ∧ T F ) ⊕ C → H ( P C , T F | P ) . Since [Λ , ξ ] = − ξ ∂∂z , we have as the image of ˜ ∇ , [ (cid:0) a + a z + a z + ( b + b z + b z ) ξ + ( c + c z + c z ) ξ (cid:1) ∂∂z ∧ ∂∂ξ , ξ ] | ξ =0 − d ∂∂z = − ( a + a z + a z ) ∂∂z − d ∂∂z = − ( a + d + a z + a z ) ∂∂z where d is a constant. Hence dim H ( F , ( ∧ T F ⊕ i ∗ ( N P C /F )) = 7 . [( − d + (1 + b + b z + b z ) ξ + ( c + c z + c z ) ξ ) ∂∂z ∧ ∂∂ξ , ξ − d ] | ξ = d = − ( b d + c d + ( b d + c d ) z + ( b d + c d ) z ) ∂∂z = ˜ ∇ (( b d + c d + ( b d + c d ) z + ( b d + c d ) z ) ∂∂z ∧ ∂∂ξ , so that obstruction vanishes and an extended Poisson analytic family V ⊂ ( F × C , ( − d − ( b d + c d ) − ( b d + c d ) z − ( b d + c d ) z + (1 + b + b z + b z ) ξ + ( c + c z + c z ) ξ ) ∂∂z ∧ ∂∂ξ ) defined by ξ = d has the characteristic map T C → H ( F , ( ∧ T F ⊕ i ∗ ( N P C /F ))( a , ..., a ) (( − a + ( a + a z + a z ) ξ + ( a + a z + a z ) ξ ) ∂∂z ∧ ∂∂ξ , a ) which is an isomorphism so that V is complete. Example 11.
Let us consider a rational ruled surface F . Let us consider the Poisson structure Λ = ξ ∂∂z ∧ ∂∂ξ . Then ξ = 0 defines a holomorphic Poisson submanifold on F which is a nonsingular rational curve ∼ = P C and the normal bundle is N P C /F ∼ = O P C ( − so that H ( P C , N P C /F ) = 0 . We compute H ( F , ( ∧ T F ⊕ i ∗ N P C /F ) • ) which is the kernel of ˜ ∇ : H ( F , ∧ T F ) → H ( P C , T F | P C ( − . Since [Λ , ξ ] = − ξ ∂∂z , we haveas the image of ˜ ∇ , [ (cid:0) a + a z + ( b + b z + b z ) ξ + ( c + c z + c z + c z ) ξ (cid:1) ∂∂z ∧ ∂∂ξ , ξ ] | ξ =0 = − ( a + a z ) ∂∂z so that H ( F , ( ∧ T F ⊕ i ∗ N P C /F ) • ) = 7 and an extended Poisson analytic family V ⊂ ( F × C , ((1 + b + b z + b z ) ξ + ( c + c z + c z + c z ) ξ ) ∂∂z ∧ ∂∂ξ ) defined by ξ = d has the characteristic map T C → H ( F , ( ∧ T F ⊕ i ∗ ( N P C /F ))( a , ..., a ) (( a + a z + a z ) ξ + ( a + a z + a z + a z ) ξ ) ∂∂z ∧ ∂∂ξ , which is an isomorphism so that V is complete. Example 12.
Let us consider a rational ruled surface F m , m ≥ . Let us consider the trivial Poissonstructure Λ = 0 on F m . Then ξ = 0 defines a holomorphic Poisson submanifold which is a nonsingularrational curve ∼ = P C and the normal bundle is N P C /F m ∼ = O P ( − m ) so that H ( P C , N P C /F m ) = 0 . Hence H ( F m , ( ∧ T F m ⊕ i ∗ N P C /F m ) • ) = H ( F m , ∧ T F m ) = k m +6 . Let us consider an extended Poisson analyticfamily V ⊂ ( F m × C m +6 , Λ( t ) = (( t + t z + t z ) ξ + ( t + t z + · · · + t m +5 z m +2 ) ξ ) ∂∂z ∧ ∂∂ξ ) defined by ξ = 0 .Then the characteristic map T C m +6 ∼ = −→ H ( F m , ( ∧ T F m ⊕ i ∗ N P C /F m ) • )( a , ..., a m +5 ) (cid:18) (( a + a z + a z ) ξ + ( a + a z + · · · + a m +5 z m +2 ) ξ ) ∂∂z ∧ ∂∂ξ , (cid:19) is an isomorphism so that V is complete at 0. Stability of compact holomorphic Poisson submanifolds
We extend the definition of a fibre manifold in [Kod63] in the context of the holomorphic Poisson category.
Definition 5.0.1.
By a holomorphic Poisson fibre manifold, we shall mean a holomorphic Poisson manifold ( W , Λ) together with a holomorphic map p of W onto a complex manifold B such that the rank of theJacobian of p at each point of W is equal to the dimension of B and Λ ∈ H ( W , ∧ T W /B ) with [Λ , Λ] = 0 .For any point u ∈ B , the inverse image p − ( u ) = ( W u , Λ u ) is a holomorphic Poisson submanifold of ( W , Λ) and call it the fibre of ( W , Λ) over u . We will denote the holomorphic Poisson fibre manifold ( W , Λ) by thequadruple ( W , Λ , B, p ) . We note that a holomorphic Poisson fibre manifold is a Poisson analytic family inthe sense of [Kim14b] when fibres are compact. For any subdomain N of B , we call the holomorphic Poissonfibre manifold ( p − ( N ) , Λ | p − ( N ) , N, p ) the restriction of ( W , Λ) to N and denote it by ( W , Λ) | N . Let V bea holomorphic Poisson submanifold of ( W , Λ) | N such that p ( V ) = N . We call V a holomorphic Poissonfibre submanifold of the holomorphic Poisson fibre manifold ( W , Λ) | N if and only if ( V , Λ | V , N, p ) forms aholomorphic Poisson fibre manifold. If, moreover, each fibre V u = V ∩ W u , u ∈ N , of V is compact, we call V a holomorphic Poisson fibre submanifold with compact fibres of the holomorphic Poisson fibre manifold ( W , Λ) | N . We extend the definition of stability in [Kod63] in the context of the holomorphic Poisson category.
Definition 5.0.2.
Let V be a compact holomorphic Poisson submanifold of a holomorphic Poisson manifold ( W, Λ ) . We call V a stable holomorphic Poisson submanifold of ( W, Λ ) if and only if, for any holomorphicPoisson fibre manifold ( W , Λ , B, p ) such that p − (0) = ( W, Λ ) for a point ∈ B , there exist a neighborhood N of in B and a holomorphic Poisson fibre submanifold V with compact fibres of the holomorphic Poissonfibre manifold ( W , Λ) | N such that V ∩ W = V . Stability of compact holomorphic Poisson submanifolds.Theorem 5.1.1 (compare [Kod63] Theorem 1) . Let V be a compact holomorphic Poisson submanifold of aholomorphic Poisson manifold ( W, Λ ) . Let N • V/W be the complex associated with the normal bundle N V/W as in Definition . . . If the first cohomology group H ( V, N • V/W ) vanishes, then V is a stable holomorphicPoisson submanifold of ( W, Λ ) . To prove Theorem 5.1.1, we extend the argument in [Kod63] p.80-85 in the context of holomorphic Poissondeformations. We tried to maintain notational consistency with [Kod63].Let ( W , Λ , B, p ) be a holomorphic Poisson fibre manifold such that p − (0) = ( W, Λ ) for a point 0 ∈ B and let ( u , ..., u q ) denote a local coordinate on B with the center 0. Considering V ⊂ W ⊂ W as asubmanifold of ( W , Λ ), we cover V by a finite number of coordinate neighborhood U i in W and choosea local coordinate ( z i , w i , u ) = ( z i , ..., z di , w i , ..., w ri , u , ..., u q ) such that the simultaneous equations w i = · · · = w ri = u = · · · = u q = 0 define V . We assume that each neighborhood U i is a polycylinder consistingof all points ( z i , w i , u ) , | z i | < , | w i | < , | u | <
1. On the intersection U i ∩ U k the local coordinates z αi , w λi are holomorphic functions of z k , w k , u : z αi = g αik ( z k , w k , u ) , α = 1 , ..., d, w λi = f λik ( z k , w k , u ) , λ = 1 , ..., r . Notethat f λik ( z k , ,
0) = 0. We set U k = V ∩ U k . We denote a point on V by z and, if z = ( z k , , ∈ U k , wecall z k = ( z k , ..., z dk ) the local coordinate of z on U k . We define a λikµ ( z ) = (cid:16) ∂f λik ( z k ,w k ,u ) ∂w µk (cid:17) w k = u =0 , b λikρ ( z ) = (cid:16) ∂f λik ( z k ,w k ,u ) ∂u ρ (cid:17) w k = w =0 . Then the normal bundle of V in W is defined by the system of transition matrices (cid:18) a ik ( z ) b iz ( z )0 1 q (cid:19) so that we have the exact sequence 0 → N V/W → N V/ W → ⊕ q O V →
0. On the otherhand, since w ii = · · · = w ri = u = · · · = u q = 0 defines a holomorphic Poisson submanifold, [Λ , w αi ] = P rβ =1 w αi T βiα ( z i , w i , u ) + P qρ =1 u ρ T q + ρiα ( z i , w i , u ) for some T γiα ( z i , w i , u ) ∈ Γ( U i , T W ), γ = 1 , ..., r + q , and[Λ , u ρ ] = 0. We note that T γiα ( z i , , ∈ Γ( U i , T W | V ). Setting T γi ( q + ρ ) ( z i , w i , u ) := 0 for ρ = 1 , ..., q, γ =1 , ..., r + q , we can write [Λ , u η ] = P rβ =1 w αi T βi ( r + η ) ( z i , w i , u ) + P qρ =1 u ρ T r + ρi ( r + η ) ( z i , w i , u ).We note that we can extend 0 → N V/W → N V/ W → ⊕ q O V → → N • V/W → R • → Q • → EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 47 · · · · · · · · · ∇ W x ∇ W x − [ − , Λ] | V x −−−−→ N V/W ⊗ ∧ T W | V −−−−→ R := N V/ W ⊗ ∧ T W | V −−−−→ Q := ⊕ q ∧ T W | V −−−−→ ∇ W x ∇ W x − [ − , Λ] | V x −−−−→ N V/W ⊗ T W | V −−−−→ R := N V/ W ⊗ T W | V −−−−→ Q := ⊕ q T W | V −−−−→ ∇ W x ∇ W x − [ − , Λ] | V x −−−−→ N V/W −−−−→ R := N V/ W −−−−→ Q := ⊕ q O V −−−−→ N • V/W is the complex associated with the normal bundle N V/W and the secondvertical complex R • is the subcomplex of the complex N • V/ W associated with the normal bundle N V/ W asin Definition 3.1.13. First we show that the second vertical complex is well-defined. Indeed, we simply notethat T βi ( r + ρ ) ( z i , ,
0) = 0 , ρ = 1 , ..., q, β = 1 , .., r + q , T βiα ( z i , , ∈ Γ( U i , T W | V ) , α = 1 , ..., r, β = 1 , ..., r + q ,and − [ g, Λ] | V ∈ Γ( U i , ∧ p +1 T W | V ) for g ∈ Γ( U i , ∧ p T W | V ).Next we show that the sequence of complex of sheaves (5.1.2) is well-defined. In other words, the abovediagram commutes. The commutativity of the first two complexes follows from the following local commu-tativity: ( − [ f i , Λ] + ( − p P rβ =1 f βi T βi ( z i ) , ..., − [ f ri , Λ] + ( − p P rβ =1 f βi T βir ( z i ) −−−−−−→ ( − [ f i , Λ] + ( − p P rβ =1 f βi T βi ( z i ) , ..., − [ f ri , Λ] + ( − p P rβ =1 f βi T βir ( z i ) , , ..., x x ( f i , ..., f ri ) −−−−−−→ ( f i , ..., f ri , , ..., where ( f i , ..., f ri ) ∈ ⊕ r Γ( U i , ∧ p T W | V ). On the other hand, the commutativity of the last two complexesfollows from the following local commutativity: ( − [ f i , Λ] | V + ( − p P r + qβ =1 h βi T βi ( z i ) , ..., − [ f ri , Λ] | V + ( − p P r + qβ =1 h βi T βir ( z i ) , − [ g i , Λ] | V , ..., − [ g qi , Λ] | V ) −−−−−−→ ( − [ g i , Λ] | V , ..., − [ g qi , Λ] | V ) x x ( h i , ..., h r + qi ) := ( f i , ..., f ri , g i , ..., g qi ) −−−−−−→ ( g i , ..., g qi ) where ( f i , ..., f ri , g i , ..., g qi ) ∈ ⊕ r + q Γ( U i , ∧ p T W | V ).Since V is compact, H ( V, Q • ) = H ( V, O qV ) ∼ = C q so that from the hypothesis H ( V, N • V/W ) = 0 and theexact sequence 0 → N • V/W → R • → Q • →
0, we obtain an exact sequence0 → H ( V, N • W/V ) → H ( V, R • ) κ −→ C q → H ( V, R • ) is a collection { ψ i ( z ) } of vector-valued holomorphic functions ψ i ( z ) = ( ψ i ( z ) , ..., ψ ri ( z ) , ψ r +1 , ..., ψ r + q )defined respectively on U i satisfying ψ λi ( z ) = P rµ =1 a λikµ ψ µk ( z ) + P qρ =1 b λikρ ( z ) ψ r + ρ and − [ ψ λi ( z ) , Λ] | V + P rβ =1 ψ βi T βiλ ( z i , ,
0) + P qρ =1 ψ r + ρ T r + ρiλ ( z i , ,
0) = 0 , λ = 1 , ..., r and we have κψ = ( ψ r +1 , ..., ψ r + q ).Let M ǫ = { t = ( t , ..., t n ) ∈ C n || t | < ǫ } , where ǫ is a small positive number. Consider a Poissonanalytic family F of compact holomorphic Poisson submanifolds V t , t ∈ M ǫ , of ( W , Λ) such that V = V (see Definition 3.0.1). Then F is a holomorphic Poisson submanifold of ( W × M ǫ , Λ) such that
F ∩ ( W × t ) = V t × t . We assume that F is covered by the neighborhoods U i × M ǫ . On each neighborhood U i × M ǫ , the holomorphic Poisson submanifold F is defined by simultaneous holomorphic equations ofthe form w λi = θ λi ( z i , t ) , λ = 1 , ..., r, u ρ = θ r + ρ ( t ) , ρ = 1 , ..., q . For any tangent vector ∂∂t = P v c v ∂∂t v of M ǫ at t = 0, we set ψ λi ( z ) = (cid:16) ∂θ λi ( z i ,t ) ∂t (cid:17) t =0 , λ = 1 , ..., r, ψ r + ρ = (cid:16) ∂θ r + ρ ( t ) ∂t (cid:17) t =0 , ρ = 1 , ..., q and let ψ i ( z ) = ( ψ i ( z ) , ..., ψ ri ( z ) , ψ r +1 , ..., ψ r + q ). Then the collection { ψ i ( z ) } of ψ i ( z ) represents an element of H ( V, N •W /V ) = H ( V, R • ) (see subsection 3.2). With this preparation, we prove Theorem 5.1.4 (compare [Kod63] Theorem 2) . There exists a Poisson analytic family F of compact holo-morphic Poisson submanifolds V t , where t ∈ M ǫ , ǫ > , of ( W , Λ) such that V = V and the characteristicmap: ∂∂t → ∂V t ∂t | t =0 maps the tangent space T ( M ǫ ) isomorphically onto H ( V, N •W /V ) = H ( V, R • ) providedthat cohomology group H ( V, N • W/V ) vanishes. Proof.
Let n = H ( V, R • ). We can choose a base { β , ..., β n } of H ( V, R • ) such that β r + ρv = 1 if v = n − q + ρ or β r + ρv = 0 otherwise for ρ = 1 , ..., q .We shall construct on each neighborhood U i × M ǫ a vector-valued holomorphic function of the form φ i ( z i , t ) = ( θ i ( z i , t ) , ..., θ ri ( z i , t ) , t n − q +1 , ..., t n ) , where t = ( t , ..., t n )satisfying the boundary conditions φ i ( z i ,
0) = 0 , (cid:18) ∂φ i ( z i , t ) ∂t v (cid:19) t =0 = β vi ( z ) , v = 1 , ..., n. such that φ i ( g ik ( z k , φ k ( z k , t )) , t ) = f ik ( z k , φ k ( z k , t )) , (5.1.5) [Λ , w αi − θ αi ( z i , t )] | w αi = θ αi ( z i ,t ) ,u ρ = t n − q + ρ = 0 , α = 1 , ..., r. (5.1.6)(Note that we have [Λ , u ρ − t n − q + ρ ] = 0 , ρ = 1 , ..., q so that we do not need to consider them.)Recall Notation 1. Then (5.1.5) and (5.1.6) are equivalent to the system of congruences φ mi ( g ik ( z k , φ mk ( z k , t )) , t ) ≡ m f ik ( z k , φ mk ( z k , t )) , m = 1 , , , · · · (5.1.7) [Λ , w αi − θ αmi ( z i , t )] | w i = θ αmi ( z i ,t ) ,u ρ = t n − q + ρ ≡ m , m = 1 , , , · · · , α = 1 , ..., r (5.1.8)As in the proof of Theorem 3 . .
1, we will construct the formal power series φ mi ( z i , t ) = ( θ mi ( z i , t ) , · · · , θ rmi ( z i , t ) , t n − q +1 , ..., t n )satisfying (5 . . m and (5 . . m by induction on m .We define φ i ( z i , t ) = P nv =1 β vi ( z ) t v . Then (5 . . holds. On the other hand, since [Λ , β αvi ( z )] | w = u =0 = P r + qβ =1 β βvi ( z ) T βiα ( z i , , , α = 1 , ..., r , [Λ , w αi − θ α i ( z i , t )] is of the form r X β =1 ( w βi − θ β i ( z i , t )) T βiα ( z i , w i , u ) + q X ρ =1 ( u ρ − t n − q + ρ ) T n − q + ρiα ( z i , w i , u ) + r X β =1 w βi P βiα ( z i , w i , u, t ) + q X ρ =1 u ρ P n − q + ρiα ( z i , w i , u, t ) , where the degree of P γiα ( z i , w i , u, t ) , γ = 1 , ..., r + q in t is 1 so that (5 . . holds. Now we assume that wehave already constructed φ mi ( z i , t ) satisfying (5 . . m and (5 . . m such that [Λ , w αi − θ αmi ( z i , t )] is of theform (as in (3.3.8))[Λ , w αi − θ αmi ( z i , t )] = r X β =1 ( w βi − θ βmi ( z i , t )) T βiα ( z i , w i , u ) + q X ρ =1 ( u ρ − t n − q + ρ ) T n − q + ρiα ( z i , w i , u ) + Q αmi ( z i , t )(5.1.9) + r X β =1 w βi P βmiα ( z i , w i , u, t ) + q X ρ =1 u ρ P ( n − q + ρ ) miα ( z i , w i , u, t ) . such that the degree of P γmiα ( z i , w i , u, t ) , γ = 1 , ..., r + q is at least 1 in t . Let ψ ik ( z, t ) ≡ m +1 φ mi ( g ik ( z k , φ mk ( z k , t ) , t ) − f ik ( z k , φ mk ( z k , t )) G αi ( z i , t ) ≡ m +1 [Λ , w αi − θ αmi ( z i , t )] | w αi = θ αmi ( z i ,t ) ,u ρ = t n − q + ρ As in the proof of Theorem 3.3.1, we can show that the collection { ψ ik ( z, t ) } ∈ C ( U ∩ V, N V/ W ) and { ( G i ( z i , t ) , ..., G ri ( z i , t ) , , ..., } ∈ C ( U ∩ V, N V/ W ⊗ T W | V ) define a 1-cocycle in the ˇCech resolution of N • V/ W .We note that { ψ ik ( z, t ) } are in C ( U ∩ V, N V/W ) and { ( G i ( z i , t ) , ..., G ri ( z i , t )) } are in C ( U ∩ V, N V/W ⊗ T W | V ) so that ( { ψ ik ( z, t ) } , { ( G i ( z i , t ) , ..., G ri ( z i , t )) } ) define an element in H ( V, N • V/W ). Since H ( V, N • V/W ) =0, there exists { χ i ( z, t ) = ( χ i ( z, t ) , ..., χ ri ( z, t )) } which are homogenous polynomials of degree m +1 in t , ..., t n EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 49 whose coefficients are in C ( U ∩ V, N V/W ) such that ψ λik ( z, t ) = r X µ =1 a λikµ ( z ) χ µk ( z, t ) − χ λi ( z, t ) , λ = 1 , ..., r. − G αi ( z i , t ) = − [ χ αi ( z i , t ) , Λ ] | w i =0 + r X β =1 χ βi ( z i , t ) T βiα ( z i , , , α = 1 , ..., r. = − [ χ αi ( z i , t ) , Λ] | w i = u =0 + r X β =1 χ βi ( z i , t ) T βiα ( z i , , φ i | m +1 ( z i , t ) = ( χ i ( z i , t ) , ..., χ ri ( z i , t ) , , ..., φ mi ( z i , t ) = φ mi ( z i , t ) + φ i | m +1 ( z i , t ) satisfy(5 . . m +1 . On the other hand, from (5.1.10), we have − [ χ αi ( z i , t ) , Λ] = − G αi ( z i , t ) − r X β =1 χ βi ( z i , t ) T βiα ( z i , w i , u ) + r X β =1 w βi R βiα ( z i , w i , u, t ) + q X ρ =1 u n − q + ρ R q + ρiα ( z i , w i , u, t )where the degree of R γiα ( z i , w i , u, t ) is m + 1 in t . Therefore we have, from (5 . . , w αi − θ αmi ( z i , t ) − χ αi ( z i , t )] = r X β =1 ( w βi − θ βmi ( z i , t ) − χ βi ( z i , t )) T βiα ( z i , w i , u ) + q X ρ =1 ( u ρ − t n − q + ρ ) T r + ρiα ( z i , w i , u ) − G αi ( z i , t ) + Q αmi ( z i , t ) + r X β =1 w βi ( P βmiα ( z i , w i , u, t ) + R βiα ( z i , w i , u, t )) + q X ρ =1 u ρ ( P ( r + ρ ) miα ( z i , w i , u, t ) + R n − q + ρiα ( z i , w i , u, t ))Lastly, we note that from (5 . . G αi ( z i , t ) ≡ m +1 Q αmi ( z i , t ) + r X β =1 θ βmi ( z i , t ) P βiα ( z i , φ mi ( z i , t ) , t ) + q X ρ =1 t n − q + ρ P ( n − q + ρ ) miα ( z i , φ mi ( z i , t ) , t )(5.1.11)so that we obtain, from (5 . .
9) and (5 . . , w αi − θ αmi ( z i , t ) − χ αi ( z i , t )] | w αi = θ αmi ( z i ,t )+ χ αi ( z i ,t ) ,u ρ = t n − q + ρ ≡ m +1 [Λ , w αi − θ αmi ( z i , t )] | w αi = θ αmi ( z i ,t )+ χ αi ( z i ,t ) ,u ρ = t n − q + ρ − [Λ , χ αi ( z i , t )] | w i = u =0 ≡ m +1 r X β =1 χ αi ( z i , t ) T βiα ( z i , ,
0) + G αi ( z i , t ) − [Λ , χ αi ( z i , t )] | w i = u =0 = 0Hence (5 . . m +1 holds. By subsection 3.4, the formal power series φ i ( z i , t ) converges for | t | < ǫ for sufficientlysmall positive number ǫ . Let M ǫ = { t = ( t , ..., t n ) ∈ C n || t | < ǫ } . Then on each neighborhood U i × M ǫ of W × M ǫ , the simultaneous equation w αi − θ αi ( z i , t ) = u ρ − t n − q + r = 0 , α = 1 , ..., r, ρ = 1 , ..., q defines thedesired Poisson analytic family F of compact holomorphic Poisson submanifolds of V t , t ∈ M ǫ of ( W , Λ) suchthat V = V . This completes the proof of Theorem 5.1.4. (cid:3) Proof of Theorem 5.1.1.
Let N ǫ = { u = ( u , ..., u q ) ∈ B || u | < ǫ } , where ǫ is a small positive number. Let F ⊂ ( W × M ǫ , Λ) be the Poisson analytic family of compact holomorphic Poisson submanifolds V t , t ∈ M ǫ of ( W , Λ) defined by w λi = θ λi ( z i , t ) , λ = 1 , , ..., r, u ρ = t n − q + ρ , ρ = 1 , , ..., q on U i × M ǫ as in the proofof Theorem 5.1.4. If we define a linear map u → t ( u ) = (0 , ..., , u , ..., u q ) of N ǫ into M ǫ , then the union V = S u V t ( u ) of the compact holomorphic Poisson submanifolds V t ( u ) , u ∈ N ǫ of ( W , Λ) which is defined by w λi = θ λi ( z i , , · · · , , u , · · · , u q ) := η λi ( z i , u , · · · , u q ) on U i forms a holomorphic Poisson fibre submanifoldwith compact fibres of the holomorphic Poisson fibre manifold ( W , Λ) | N ǫ with V ∩ W = V so that V is astable holomorphic Poisson submanifold of ( W, Λ ). (cid:3) As in Theorem 3 in [Kod63], by combining the exact sequence (5.1.3) and Theorem 5.1.1, we can show
Theorem 5.1.12.
Let ( W , Λ , B, p ) be a holomorphic Poisson fibre manifold such that ( W, Λ ) = p − (0) isthe fibre of ( W , Λ) over a point ∈ B . Let V be a compact holomorphic Poisson submanifold of ( W, Λ ) .Assume that H ( V, N • W/V ) = H ( V, N • W/V ) = 0 . Then, for a sufficiently small neighborhood N of ∈ B , there exists the unique holomorphic Poisson fibre submanifold V with compact fibers of the holomorphicPoisson fibre manifold ( W , Λ) | N such that V ∩ W = V . Example 13.
We keep the notations in Example . Let us consider a Poisson rational ruled surface ( F , z ξ ∂∂z ∧ ∂∂ξ ) . We show that the holomorphic Poisson submanifold ξ = ξ ′ = 0 of ( F , z ξ ∂∂z ∧ ∂∂ξ ) isunstable. Take two copies of U i × P C × C , where U i = C and write the coordinates as ( z, [ ξ , ξ ] , t ) and ( z ′ , [ ξ ′ , ξ ′ ] , t ′ ) . Patch U i × P C × C , i = 1 , by the relation z ′ = z , t = t ′ and [ ξ ′ , ξ ′ ] = [ ξ , z ξ + tzξ ] anddenote it by X . Then the projection π : X → C define a complex analytic family of deformations of F . Wegive a holomorphic Poisson structure on X to make a Poisson analytic family. We set ξ = ξ ξ and ξ ′ = ξ ′ ξ ′ .Since − ξ ′ ∂∂z ′ ∧ ∂∂ξ ′ = ( z ξ + tz ) ∂∂z ∧ ∂∂ξ , π : ( X, Λ = ( z ξ + tz ) ∂∂z ∧ ∂∂ξ ) → C defines a Poisson analytic family.Since ξ = ξ ′ = 0 can not be extended to a complex analytic family as in [Kod63] p.86, it can not be extendedto a Poisson analytic family as a holomorphic Poisson fibre submanifold of ( X, Λ) so that it is not stable. Example 14.
Let us consider F ∼ = P C × P C and a Poisson structure Λ = ξ ∂∂z ∧ ∂∂ξ on F . We keep thenotations in Example . We show that the holomorphic Poisson submanifold V : ξ = 0 is unstable. Let usconsider a Poisson analytic family ( F × C , Λ = ( ξ − tz ) ∂∂z ∧ ∂∂ξ ) . Assume that there is a holomorphic Poissonfibre manifold of V ⊂ ( F × B, Λ) , where B = { t ∈ C || t | < ǫ } for a sufficiently small number ǫ > such that V| t =0 is ξ = ξ ′ = 0 . We may assume that V is defined by ξ − ϕ ( z, t ) = 0 on U × P C × C and ξ ′ − ϕ ( z ′ , t ) = 0 on U × P C × C , where U i = C , i = 1 , so that we have a relation ϕ ( z, t ) − ϕ ( z , t ) = 0 . Hence ϕ i ( z, t ) is ofthe from ϕ ( z, t ) = f ( t ) . On the other hand, since ξ − ϕ ( z, t ) defines a holomorphic Poisson submanifold of F × C , [Λ , ξ − ϕ ( z, t )] | ξ = ϕ ( z,t ) = 0 so that − ( ξ − tz ) ∂∂z | ξ − ϕ ( z,t ) = 0 ⇐⇒ ϕ ( z, t ) = tz which contradictsto ϕ ( z, t ) = f ( t ) . Hence V : ξ = 0 is not stable as a holomorphic Poisson submanifold while it is stable asa complex submanifold since H ( V, N V/F ) ∼ = H ( P C , O P C ) = 0 . Appendix A. Deformations of Poisson structures
We denote by
Art the category of local artinian k -algebras with residue field k , where k is an algebraicallyclosed field with characteristic 0, and by k [ ǫ ] by the ring of dual numbers. Definition A.0.13.
Let ( Y, Λ ) be a nonsingular Poisson variety. An infinitesimal deformation of Λ over A ∈ Art is an algebraic Poisson scheme ( Y × Spec ( k ) A, Λ) which induces ( Y, Λ ) , where Λ ∈ H ( Y, ∧ T Y ) ⊗ A .Then for each A ∈ Art , we can define a functor of Artin rings
Def Λ : Art → ( sets ) A
7→ { infinitesimal deformations of Λ over A } We will denote by H i ( Y, ∧ T •− Y ) the i -th hypercohomology group of the following complex of sheaves ∧ T •− Y : ∧ T Y − [ − , Λ ] −−−−−→ ∧ T Y − [ − , Λ ] −−−−−→ ∧ T Y − [ − , Λ ] −−−−−→ · · · (A.0.14) Proposition A.0.15.
Let ( Y, Λ ) be a nonsingular Poisson variety. Then (1) There is a natural identification
Def Λ ( k [ ǫ ]) ∼ = H ( Y, ∧ T •− Y ) . (2) Given an infinitesimal deformation η of Λ over A ∈ Art and a small extension → ( t ) → ˜ A → A → , we can associate an element o η ( e ) ∈ H ( Y, ∧ T •− Y ) , which is zero if and only if there is alifting of η to ˜ A .Proof. Let Λ ∈ H ( Y, ∧ T Y ) ⊗ k [ ǫ ] be an infinitesimal deformation of Λ over Spec ( k [ ǫ ]) so that Λ = Λ + ǫ Λ ′ for some Λ ′ ∈ H ( Y, ∧ T Y ). Since [Λ + ǫ Λ ′ , Λ + ǫ Λ ′ ] = 0 so that − [Λ ′ , Λ ] = 0. Hence Λ ′ ∈ H ( Y, ∧ T •− Y ).Now we identify obstructions. Consider a small extension e : 0 → ( t ) → ˜ A → A →
0. Let η bean infinitesimal deformations of Λ over A , in other words, a Poisson structure Λ ∈ H ( Y, ∧ T Y ) ⊗ A on Y × Spec ( k ) A over A . Let U = { U i } be an affine open covering of Y . Then Λ is locally expressed asΛ i ∈ Γ( U i , ∧ T Y ) ⊗ A with [Λ i , Λ i ] = 0. Let ˜Λ i ∈ H ( Y, ∧ T Y ) ⊗ ˜ A be an arbitrary lifting of Λ i to ˜ A so that EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 51 [˜Λ i , ˜Λ i ] = t Π i for some Π i ∈ Γ( U i , ∧ T Y ) and ˜Λ i − ˜Λ j = t Λ ′ ij for some Λ ′ ij ∈ Γ( U i ∩ U j , ∧ T Y ). Then we have t [Λ , Π i ] = [˜Λ i , [˜Λ i , ˜Λ i ]] = 0 ⇐⇒ − [ 12 Π i , Λ ] = 0(A.0.16) ( 12 Π i −
12 Π j ) = 12 [˜Λ i , ˜Λ i ] −
12 [˜Λ j , ˜Λ j ] = t [Λ , Λ ′ ij ] ⇐⇒ δ ( 12 Π i ) − [ − Λ ′ ij , Λ ] = 0(A.0.17) t (Λ ′ jk − Λ ′ ik + Λ ′ ij ) = ˜Λ j − ˜Λ k − ˜Λ i + ˜Λ k + ˜Λ i − ˜Λ j = 0 ⇐⇒ − δ ( − Λ ′ ij ) = 0 . (A.0.18)Hence ( { Π i } , {− Λ ′ ij } ) ∈ C ( U , ∧ T Y ) ⊕ C ( U , ∧ T Y ) define a 1-cocycle in the following ˇCech resolution of ∧ T •− X : C ( U , ∧ T Y ) − [ − , Λ ] x C ( U , ∧ T Y ) δ −−−−→ C ( U , ∧ T Y ) − [ − , Λ ] x − [ − , Λ ] x C ( U , ∧ T Y ) − δ −−−−→ C ( U , ∧ T Y ) δ −−−−→ C ( U , ∧ T Y )Now we choose another arbitrary lifting ˜Λ ′ i ∈ Γ( U i , ∧ T Y ) ⊗ ˜ A of Λ i . We show that the associated cohomologyclass b := ( { Π ′ i } , {− Λ ′′ ij } ) is cohomologous to a := ( { Π i } , {− Λ ′ ij } ). We note that ˜Λ ′ i = ˜Λ i + tD i for some D i ∈ Γ( U i , ∧ T Y ). Then t
12 Π ′ i − t
12 Π i = 12 [˜Λ ′ i , ˜Λ ′ i ] −
12 [˜Λ i , ˜Λ i ] = [ tD i , Λ ] ⇐⇒
12 Π i −
12 Π ′ = − [ D i , Λ ](A.0.19) t Λ ′′ ij − t Λ ′ ij = ˜Λ ′ i − ˜Λ ′ j − ˜Λ i + ˜Λ j = t ( D i − D j ) ⇐⇒ − Λ ′ ij − ( − Λ ′′ ij ) = − δ ( D i )(A.0.20)Hence { D i } ∈ C ( U , ∧ T Y ) is mapped to a − b so that a is cohomologous to b . So given a small extension e : 0 → ( t ) → ˜ A → A →
0, we can associate an element o η ( e ) := the cohomology class of a ∈ H ( Y, ∧ T •− Y ).We note that o η ( e ) = 0 if and only if there exists a collection { ˜Λ i } such that Π i = 0 (which means [˜Λ i , ˜Λ i ] = 0)and Λ ′ ij = 0 (which means { ˜Λ i } glues together to define a Poisson structure on Y × Spec ( ˜ A )) if and only ifthere is a lifting of η to ˜ A . (cid:3) Remark A.0.21.
We have an exact sequence of complex of sheaves → ∧ T •− X → T • X → T X → · · · · · · · · · − [ − . Λ ] x − [ − , Λ ] x x −−−−→ ∧ T X −−−−→ ∧ T X −−−−→ −−−−→ − [ − , Λ ] x − [ − , Λ ] x x −−−−→ ∧ T X −−−−→ ∧ T X −−−−→ −−−−→ x − [ − , Λ ] x x −−−−→ −−−−→ T X −−−−→ T X −−−−→ which induces H ( X, ∧ T •− X ) → H ( X, T • X ) → H ( X, T X )(A.0.22) H ( X, ∧ T •− X ) → H ( X, T • X ) → H ( X, T X )(A.0.23) We also have morphisms of deformation functors
Def Λ → Def ( X, Λ ) → Def X (A.0.24) where Def ( X, Λ ) is the functor of flat Poisson deformations of ( X, Λ ) ( see [Kim14a]) and Def X is thefunctor of flat deformations of X ( see [Ser06] p. . Then ( A. . represents the morphisms of tangent spaces for ( A. .
24) :
Def Λ ( k [ ǫ ]) → Def ( X, Λ ) ( k [ ǫ ]) → Def X ( k [ ǫ ]) , and ( A. . represents the obstructionmaps for ( A. . . Appendix B. Deformations of Poisson closed subschemes
B.1.
The local Poisson Hilbert functor.
Let X ⊂ ( Y, Λ ) be a closed embedding of algebraic Poisson schemes, where ( Y, Λ ) is a nonsingularPoisson variety. An infinitesimal deformation of X in ( Y, Λ ) over A ∈ Art is a cartesian diagram ofmorphisms of schemes X −−−−→ X ⊂ ( Y × Spec ( A ) , Λ ) y y π Spec ( k ) −−−−→ S = Spec ( A )where π is flat and induced by a projection from Y × S to S , and X is a Poisson closed subscheme of( Y × S, Λ ). Then we can define a functor of Artin rings (called the local Poisson Hilbert functor of X in( Y, Λ )) H ( Y, Λ ) X : Art → ( Sets ) A
7→ { infinitesimal deformations of X in ( Y, Λ ) over A } B.2.
The complex associated with the normal bundle of a Poisson closed subscheme of a non-singular Poisson variety.
Let ( Y, Λ ) be a nonsingular Poisson variety and X be a Poisson closed subscheme of ( Y, Λ) defined by aPoisson ideal sheaf I . Assume that i : X ֒ → Y be a regular embedding. Let { U i } be an affine open cover of Y such that I i = ( f i , ..., f Ni ) be a Poisson ideal of Γ( U i , O Y ) defining X ∩ U i and { f i , ..., f Ni } is a regularsequence. Since ( f i , ..., f Ni ) = ( f j , ..., f Nj ), f αi = P Nβ =1 r αijβ f βj for some r αijβ ∈ Γ( U i ∩ U j , O Y ). Since I i /I i is free Γ( U i ∩ X, O X )-module and generated by { f αi + I i } , α = 1 , ..., N , ¯ r βijα is uniquely determined, where¯ r αijβ is the restriction of r αijβ to Γ( U i ∩ U j ∩ X, O X ). Then the normal sheaf N X/Y := H om O X ( I / I , O X )is locally described in the following way: Hom Γ( U i ∩ X, O X ) ( I i /I i , Γ( U i ∩ X, O X ) ∼ = ⊕ N Γ( U i ∩ X, O X ) , φ ( φ ( ¯ f i ) , ..., φ ( ¯ f Ni )), where ¯ f αi is the image of f αi ∈ I i in I i /I i , and on U i ∩ U j , ( g j , ..., g Nj ) ∈ ⊕ N Γ( U j ∩ X, O X )is identified with ( P Nβ =1 ¯ r ijβ g βj , ..., P Nβ =1 ¯ r Nijβ g βj ) ∈ ⊕ N O X ( U i ∩ X ).On the other hand, ( f i , ..., f Ni ) is a Poisson ideal, in other words, { f αi , O Y } ⊂ ( f i , ..., f Ni ) , α = 1 , ..., N so that [Λ , f αi ] = P nβ =1 f βi T βiα for some T βiα ∈ Γ( U i , T Y ). Let ¯ T βiα be the image of T βiα in Γ( U i ∩ X, T Y | X ).Then we have(1) We note that P Nβ,γ =1 r βijγ f γj T βiα = P Nβ =1 f βi T βiα = [Λ , f αi ] = [Λ , P Nβ =1 r αijβ f βj ] = P Nβ =1 [Λ , r αijβ ] f βj + P Nβ =1 r αijβ [Λ , f βj ] = P Nβ =1 [Λ , r αijβ ] f βj + P Nβ,γ =1 r αijβ f γj T γjβ . Then P Nγ =1 f γj ( P Nβ =1 r βijγ T βiα − [Λ , r αijγ ] − P Nβ =1 r αijβ T γjβ ) = 0 so that we get N X β =1 ¯ r βijγ ¯ T βiα = [Λ , r αijγ ] + N X β =1 ¯ r αijβ ¯ T γjβ (B.2.1) where [Λ , r αijγ ] is the image of [Λ , r αijγ ] in Γ( U i ∩ X, T Y | X ).(2) By taking [Λ , − ] on [Λ , f αi ] = P Nβ =1 f βi T βiα , we get P Nβ =1 f βi [Λ , T βiα ] − P Nβ =1 [Λ , f βi ] ∧ T βiα = 0.Then P Nγ =1 f γi [Λ , T γiα ] − P Nβ,γ =1 f γi T γiβ ∧ T βiα = P Nγ =1 f γi ([Λ , T γiα ] − P Nβ =1 T γiβ ∧ T βiα ) = 0 so that weget [Λ , T γiα ] − N X β =1 ¯ T γiβ ∧ ¯ T βiα = 0 , (B.2.2) where [Λ , T γiα ] is the image of [Λ , T γiα ] in Γ( U i ∩ X, ∧ T Y | X ).Now we define the complex N • X/Y associated with the normal bundle N X/Y : N • X/Y : N X/Y ∇ −→ N X/Y ⊗ T Y | X ∇ −→ N X/Y ⊗ ∧ T Y | X ∇ −→ · · · (B.2.3) EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 53
The complex is defined locally in the following way: ∇ : ⊕ N Γ( U i ∩ X, ∧ p T Y | X ) → ⊕ N Γ( U i ∩ X, ∧ p +1 T Y | X )( g i , ..., g Ni ) ( − [ g i , Λ ] + ( − p N X β =1 g βi ∧ ¯ T βi , ..., − [ g Ni , Λ ] + ( − p N X β =1 g βi ∧ ¯ T βiN )We denote the i -th hypercohomology group of N • X/Y by H i ( X, N • X/Y ). Proposition B.2.4 (compare [Ser06] Proposition 3 . . . . . Given a regular closed em-bedding of algebraic Poisson schemes i : X ֒ → ( Y, Λ ) , where ( Y, Λ ) is a nonsingular Poisson variety,then (1) There is a natural identification H ( Y, Λ ) X ( k [ ǫ ]) ∼ = H ( X, N • X/Y )(2)
Given an infinitesimal deformation η of X in ( Y, Λ ) over A ∈ Art and a small extension e : 0 → ( t ) → ˜ A → A → , we can associate an element o η ( e ) ∈ H ( X, N • X/Y ) , which is zero if and only ifthere is a lifting of η to ˜ A .Proof. Let U = { U i } be an affine open covering of Y and let I i = ( f i , ..., f Ni ) be a Poisson ideal of Γ( U i , O Y )defining U i ∩ X such that { f i , · · · f Ni } is a regular sequence. We keep the notations in subsection B.2.A first-order deformation of X in ( Y, Λ ) is a flat family X −−−−→ X ⊂ ( Y × Spec ( k [ ǫ ]) , Λ ) y y Spec ( k ) −−−−→ Spec ( k [ ǫ ])so that X is determined by a Poisson ideal sheaf I generated by { f αi + ǫg αi } , α = 1 , ..., N for some g αi ∈ Γ( U i , O Y ). Let (¯ g i , ..., ¯ g Ni ) ∈ ⊕ N Γ( X ∩ U i , O X ) be the image of ( g i , ..., g Ni ) ∈ ⊕ N Γ( U i , O Y ). Since ( f i + ǫg i , ..., f Ni + ǫg Ni ) = ( f j + ǫg j , ..., f Ni + ǫg Ni ), f αi + ǫg αi = P Nβ =1 ( r αijβ + ǫh αijβ )( f βj + ǫg βj ) for some h αijβ ∈ Γ( U i ∩ U j , O Y ) so that we have ¯ g αi = P nβ =1 ¯ r αijβ ¯ g βi . Hence { (¯ g i , ..., ¯ g Ni ) } define a global section in H ( X, N Y/X ).On the other hand, since ( f i + ǫg i , ..., f Ni + ǫg Ni ) is a Poisson ideal, we have [Λ , f αi + ǫg αi ] = P Nβ =1 ( f βi + ǫg βi )( T βiα + ǫW βiα ) for some W βiα ∈ Γ( U i , T Y ). Then [Λ , g αi ] = P Nβ =1 ¯ g βi ¯ T βiα so that ∇ ( { (¯ g i , ..., ¯ g Ni ) } ) = 0.Hence { (¯ g i , ..., ¯ g Ni ) } ∈ H ( X, N • X/Y ).Now we identify obstructions. Consider a small extension e : 0 → ˜ A → A →
0. Let η := ( X ⊂ ( Y × Spec ( A ) , Λ )) be an infinitesimal deformation of X in ( Y, Λ ) over A . Then X is determined by aPoisson ideal sheaf I A generated by ( F i , ..., F Ni ) in Γ( U i , O Y ) ⊗ A such that F αi ≡ f αi ⊗ m A ) and { F i , ..., F Ni } is a regular sequence. Since ( F i , ..., F Ni ) = ( F j , ..., F Nj ), we have F αi = P Nβ =1 R αijβ F βj forsome R αijβ ∈ Γ( U i , O Y ) ⊗ A . On the other hand, since ( F i , ..., F Ni ) is a Poisson ideal, we have [Λ , F αi ] = P Nβ =1 F βi W βiα for some W βiα ∈ Γ( U i , T Y ) ⊗ A . Let ˜ F αi ∈ Γ( U i , O Y ) ⊗ A be an arbitrary lifting of F αi ,˜ T βiα ∈ Γ( U i , T Y ) ⊗ ˜ A be an arbitrary lifting of W βiα , and ˜ R βijα ∈ Γ( U i ∩ U j , O Y ) ⊗ ˜ A be an arbitrary liftingof R αijβ . Then [Λ , ˜ F αi ] − P Nβ =1 ˜ F βi ˜ T βiα = tG αi for some G αi ∈ Γ( U i , T Y ), ˜ F αi − P Nβ =1 ˜ R αijβ ˜ F βj = th αij for some h αij ∈ Γ( U i ∩ U j , O Y ), and ˜ R αikγ − P Nβ =1 ˜ R αijβ ˜ R βjkγ = tP αijkγ for some P αijkγ ∈ Γ( U i ∩ U j ∩ U k , O Y ). We willshow that { ( − ¯ G i , ..., − ¯ G Ni ) } ⊕ { (¯ h ij , ..., ¯ h Nij ) } ∈ C ( U , N X/Y ⊗ T Y | X ) ⊕ C ( U , N X/Y ) define a 1-cocycle inthe following ˇCech resolution of N • X/Y : C ( U ∩ X, N X/Y ⊗ ∧ T Y | X ) ∇ x C ( U ∩ X, N X/Y ⊗ T Y | X ) δ −−−−→ C ( U ∩ X, N X/Y ⊗ T Y | X ) ∇ x ∇ x C ( U ∩ X, N X/Y ) − δ −−−−→ C ( U ∩ X, N X/Y ) δ −−−−→ C ( U ∩ X, N X/Y )First we show that ∇ ( { ( − ¯ G i , ..., − ¯ G Ni ) } ) = 0. As in (B.2.2), we can show P Nγ =1 F γi ([Λ , W γiα ] − P Nβ =1 W γiβ ∧ W βiα ) = 0 so that we have P Nγ =1 ˜ F γi ([Λ , ˜ T γiα ] − P Nβ =1 ˜ T γiβ ∧ ˜ T βiα ) = P Nγ =1 ˜ F γi tQ γiα = P Nγ =1 f γi tQ γiα for some Q γiα ∈ Γ( U i , ∧ T Y ). Then we have t [Λ , G αi ] = − N X β =1 [Λ , ˜ F βi ˜ T βiα ] = − N X β =1 ˜ F βi [Λ , ˜ T βiα ] + N X β =1 [Λ , ˜ F βi ] ∧ ˜ T βiα (B.2.5)= − N X γ =1 ˜ F γi [Λ , ˜ T γiα ] + N X β =1 tG βi ∧ T βiα + N X β,γ =1 ˜ F γi ˜ T γiβ ∧ ˜ T βiα = − N X γ =1 ˜ F γi ([Λ , ˜ T γiα ] − ˜ T γiβ ∧ ˜ T βiα ) + N X β =1 tG βi ∧ T βiα By taking − on (B.2.5), we obtain t [Λ , G αi ] = − N X γ =1 ¯ f γi t ¯ Q γiα + N X β =1 t ¯ G βi ∧ ¯ T βiα = N X β =1 t ¯ G βi ∧ ¯ T βiα ⇐⇒ − [ − G αi , Λ ] + ( − N X β =1 − ¯ G βi ∧ ¯ T βiα = 0(B.2.6)Next we show that δ ( { ( − ¯ G i , ..., − ¯ G Ni ) } ) + ∇ ( { ¯ h ij , ..., ¯ h Nij } ) = 0. We have t ( G αi − N X β =1 r αijβ G βj ) = [Λ , ˜ F αi ] − N X β =1 ˜ F βi ˜ T βiα − N X β =1 ˜ R αijβ [Λ , ˜ F βj ] + N X β,γ =1 ˜ R αijβ ˜ F γj ˜ T γjβ (B.2.7)On the other hand, we have t [Λ , h αij ] − t N X β =1 h βij T βiα = [Λ , ˜ F αi ] − N X β =1 [Λ , ˜ R αijβ ˜ F βj ] − N X β =1 ˜ F βi ˜ T βiα + N X β,γ =1 ˜ R βijγ ˜ F γj ˜ T βiα (B.2.8) = [Λ , ˜ F αi ] − N X β =1 ˜ F βj [Λ , ˜ R αijβ ] − N X β =1 ˜ R αijβ [Λ , ˜ F βj ] − N X β =1 ˜ F βi ˜ T βiα + N X β,γ =1 ˜ R βijγ ˜ F γj ˜ T βiα As in (B.2.1), we can show P Nγ =1 F γj ( P Nβ =1 R βijγ W βiα − [Λ , R αijγ ] − P Nβ =1 R αijβ W γjβ ) = 0 so that we have P Nγ =1 ˜ F γj ( P Nβ =1 ˜ R βijγ ˜ T βiα − [Λ , ˜ R αijγ ] − P Nβ =1 ˜ R αijβ ˜ T γjβ ) = P Nγ =1 ˜ F γj tS αijγ = P Nγ =1 f γj tS αijγ for some S αijγ ∈ Γ( U i ∩ U j , T Y ). Then from (B.2.7) and (B.2.8), we obtain t ( G αi − N X β =1 r αijβ G βj ) − t ([Λ , h αij ] − N X β =1 T βiα h βij ) = N X β,γ =1 ˜ F γj ˜ R αijβ ˜ T γjβ + N X β =1 ˜ F γj [Λ , ˜ R αijγ ] − N X β,γ =1 ˜ F γj ˜ R βijγ ˜ T βiα (B.2.9)By taking − on (B.2.9) , we get t ( ¯ G αi − N X β =1 ¯ r αijβ ¯ G βj ) − t ([Λ , h αij ] − N X β =1 ¯ h βij ¯ T βiα ) = − N X γ =1 ¯ f γi t ¯ S αijγ = 0 ⇐⇒ ( ¯ G αi − N X β =1 ¯ r αijβ ¯ G βj ) + ( − [ h αij , Λ ] + N X β =1 ¯ h βij ¯ T βiα ) = 0(B.2.10) EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 55
Lastly, we show that δ ( { (¯ h ij , ..., ¯ h Nij ) } ) = 0. t ( N X β =1 r αijβ h βjk − h αik + h αij ) = N X β =1 ˜ R αijβ ˜ F βj − N X β,γ =1 ˜ R αijβ ˜ R βjkγ ˜ F γk − ˜ F αi + N X β =1 ˜ R αikβ ˜ F βk + ˜ F αi − N X β =1 ˜ R αijβ ˜ F βj (B.2.11)= − N X β,γ =1 ˜ R αijβ ˜ R βjkγ ˜ F γk + N X β =1 ˜ R αikβ ˜ F βk = N X γ =1 ( ˜ R αikγ − N X β =1 ˜ R αijβ ˜ R βjkγ ) ˜ F γk = N X γ =1 tP αijkγ ˜ F γk = N X γ =1 tP αijkγ f kγ By taking − on (B.2.11), we get t ( N X β =1 ¯ r αijβ ¯ h βjk − ¯ h αik + ¯ h αij ) = N X γ =1 t ¯ P αijkγ ¯ f kγ = 0 ⇐⇒ N X β =1 ¯ r αijβ ¯ h βjk − ¯ h αik + ¯ h αij = 0(B.2.12)Hence from (B.2.13), (B.2.10) and (B.2.12), { ( − ¯ G i , ..., − ¯ G Ni ) } ⊕ { (¯ h ij , ..., ¯ h Nij ) } ∈ C ( U ∩ X, N X/Y ⊗ T Y | X ) ⊕ C ( U ∩ X, N X/Y ) define a 1-cocycle in the above ˇCech resoultion.Now we choose another arbitrary lifting F ′ αi ∈ Γ( U i , O Y ) ⊗ ˜ A of F αi , another arbitrary lifting ˜ T ′ βiα ∈ Γ( U i , T Y ) ⊗ ˜ A of W βiα and another arbitrary lifting R ′ αijβ ∈ Γ( U i ∩ U j , O Y ) ⊗ ˜ A . We show that the associatedcohomology class b := { ( − ¯ G ′ i , ..., − ¯ G ′ Ni ) } ⊕ { (¯ h ′ ij , ..., ¯ h ′ Nij ) } is cohomologous to a := { ( − ¯ G i , ..., − ¯ G Ni ) } ⊕{ (¯ h ij , ..., ¯ h Nij ) } . We note that ˜ F ′ αi = ˜ F αi + tA αi for some A αi ∈ Γ( U i , O Y ), ˜ T ′ βiα = ˜ T βiα + tB βiα for some B βiα ∈ Γ( U i , T Y ) and ˜ R ′ αijβ = ˜ R αijβ + tC αijβ for some C αijβ ∈ Γ( U i ∩ U j , O Y ). t ( G ′ αi − G αi ) = [Λ , ˜ F ′ αi ] − N X β =1 ˜ F ′ βi ˜ T ′ βiα − [Λ , ˜ F αi ] + N X β =1 ˜ F βi ˜ T βiα = [Λ , tA αi ] − N X β =1 tA βi T βiα − N X β =1 f βi tB βiα (B.2.13)By taking − on (B.2.13), we get − ¯ G ′ αi − ( − ¯ G αi ) = − [ ¯ A αi , Λ ] + P Nβ =1 ¯ A βi ¯ T βiα .On the other hand, t ( h ′ αij − h αij ) = ˜ F ′ αi − N X β =1 ˜ R ′ αijβ ˜ F ′ βj − ˜ F αi + N X β =1 ˜ R αijβ ˜ F βj = tA αi − N X β =1 r αijβ tA βj − N X β =1 tC αijβ f βj . (B.2.14)By taking − on (B.2.14), we get h ′ αij − h αij = ¯ A αi − P Nβ =1 ¯ r αijβ ¯ A βj . Hence { ( ¯ A i , ..., ¯ A Ni ) } is mapped to b − a so that a is cohomologous to a . So given a small extension e : 0 → ( t ) → ˜ A → A →
0, we can associate anelement o η ( e ) := the cohomology class a ∈ H ( X, N • X/Y ). We note that o η ( e ) = 0 if and only if there existsa collection { ˜ F αi } , { ˜ T βiα } and { ˜ R αijβ } such that ¯ h αij = 0 and ¯ G αi = 0 , α = 1 , ..., N :(1) If ¯ h αij = 0, then h αij = f j L j + · · · f Nj L Nj for some L βj ∈ Γ( U i ∩ U j , O Y ). Then ˜ F αi − P Nβ =1 ˜ R αijβ ˜ F βj = t ( P Nβ =1 ˜ F βj L βj ). Hence ˜ F αi − P Nβ =1 ( ˜ R αijβ + tL βj ) ˜ F βj = 0 so that ( ˜ F i , ..., ˜ F Ni ) = ( ˜ F j , ..., ˜ F Nj ). Hence { ( ˜ F i , ..., ˜ F Ni ) } is an ideal sheaf on Y × Spec ( ˜ A ).(2) If ¯ G αi = 0, then G αi = f i P i + · · · + f Ni P Ni for some P βi ∈ Γ( U i , T Y ), then [Λ , ˜ F αi ] − P Nβ =1 ˜ F βi ˜ T βiα = t P Nβ =1 f βi P βi = t P Nβ =1 ˜ F βi P βi . Hence [Λ , ˜ F αi ] = P Nβ =1 ˜ F βi ( ˜ T βiα + tP iβ ) so that ( ˜ F i , ..., ˜ F Ni ) is aPoisson ideal.Hence o η ( e ) = 0 if and only if there is a lifting of η to ˜ A . (cid:3) B.3.
Deformations of Poisson closed subschemes of codimension and Poisson semi-regularity. (compare [Ser06] p.143-144)Let ( L, ∇ ) be a Poisson invertible sheaf on a nonsingular Poisson variety ( X, Λ ), where ∇ is a Poissonconnection on L , and we denote by H i ( X, L • ) the i -th hypercohomology group of the following complex ofsheaves (see [Kim14a]) L • : L ∇ −→ L ⊗ T X ∇ −→ L ⊗ ∧ T X ∇ −→ L ⊗ ∧ T X ∇ −→ · · · (B.3.1) Let s ∈ H ( X, L • ) and D = div ( s ) ⊂ ( X, Λ ) be the Poisson divisor associated with s . Then we have amorphism of functors of Artin rings: H ( X, Λ ) D → Def ( L, ∇ ) (B.3.2)where Def ( L, ∇ ) is the functor of deformations of ( L, ∇ ) (see [Kim14a]). Let us consider the following exactsequence 0 → O • X s −→ L • → L • D → · · · · · · · · · − [ − , Λ ] x ∇ x x −−−−→ ∧ T X s −−−−→ L ⊗ ∧ T X −−−−→ L D ⊗ ∧ T X | D −−−−→ − [ − , Λ ] x ∇ x x −−−−→ T X s −−−−→ L ⊗ T X −−−−→ L D ⊗ T X | D −−−−→ − [ − , Λ ] x ∇ x x −−−−→ O X s −−−−→ L −−−−→ L D −−−−→ → H ( X, O • X ) → H ( X, L • ) → H ( D, L • D ) δ −→ H ( X, O • X ) → H ( X, L • ) → H ( D, L • D ) δ −→ H ( X, O • X ) → · · · Then δ represents the morphism of tangent spaces for (B.3.2): H ( X,A ) D ( k [ ǫ ]) → Def ( L, ∇ ) ( k [ ǫ ]) and δ represents the obstruction map for (B.3.2). Remark B.3.3 (Poisson semi-regularity) . Let ( X, Λ ) be a nonsingular Poisson projective variety. A Pois-son Cartier divisor D on X is called Poisson semi-regular if the natural map H ( X, O X ( D ) • ) → H ( D, O D ( D ) • ) is zero. If D ⊂ ( X, Λ ) is Poisson semi-regular and Def ( L, ∇ ) is smooth, then H ( X, Λ ) D is smooth so that D ⊂ ( X, Λ ) is unobstructed. Appendix C. Simultaneous deformations of Poisson structures and Poisson closedsubschemes
C.1.
The local extended Poisson Hilbert functor.
Let X ⊂ ( Y, Λ ) be a closed embedding of algebraic Poisson schemes where ( Y, Λ ) is a nonsingularPoisson variety, and A ∈ Art . An infinitesimal simultaneous deformation of X in ( Y, Λ ) over A ∈ Art is acartesian diagram of morphisms of schemes η : X −−−−→ X ⊂ ( Y × Spec ( A ) , Λ) y y π Spec ( k ) −−−−→ S = Spec ( A )where π is flat, and it is induced by the projection from Y × S to S , Λ ∈ Γ( Y × S, H om ( ∧ Ω Y × S/S , O Y × S ))defines a Poisson structure on Y × S over S , X is a Poisson closed subscheme of ( Y × S, Λ), and ( Y × S, Λ)induces ( Y, Λ ). Then we can define a functor of Artin rings (called the local extended Poisson Hilbertfunctor of X in ( Y, Λ )) EH ( Y, Λ ) X : Art → ( Sets ) A
7→ { infinitesimal simultaneous deformations of X in ( Y, Λ ) } C.2.
The extended complex associated with the normal bundle N X/Y of a Poisson closed sub-schemes of a nonsingular Poisson variety.
Let ( Y, Λ ) be a nonsingular Poisson variety and X be a Poisson closed subscheme of ( Y, Λ ) defined by aPoisson ideal sheaf I . Assume that i : X ֒ → Y be a regular embedding. We keep the notations in subsectionB.2. EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 57
We define the extended complex ( ∧ T Y ⊕ i ∗ N X/Y ) • associated with the normal bundle N X/Y :( ∧ T Y ⊕ i ∗ N X/Y ) • : ∧ T Y ⊕ i ∗ N X/Y ˜ ∇ −→ ∧ T Y ⊕ ( i ∗ N X/Y ⊗ T Y | X ) ˜ ∇ −→ ∧ T Y ⊕ ( i ∗ N X/Y ⊗ ∧ T Y | X ) ˜ ∇ −→ · · · (C.2.1)The complex is defined locally in the following way:˜ ∇ : Γ( U i , ∧ p +2 T Y ) ⊕ ( ⊕ r Γ( U i ∩ X, ∧ p T Y | X ) → Γ( U i , ∧ p +3 T Y ) ⊕ ( ⊕ r Γ( U i ∩ X, ∧ p +1 T Y | X )(Π i , ( g i , ..., g Ni )) ( − [Π i , Λ ] , [Π i , f i ] − [ g i , Λ ] + ( − p N X β =1 g βi ∧ ¯ T βi , ..., [Π i , f Ni ] − [ g Ni , Λ ] + ( − p N X β =1 g βi ∧ ¯ T βiN ) Proposition C.2.2.
Given a regular closed embedding of algebraic Poisson schemes i : X ֒ → ( Y, Λ ) , where ( Y, Λ ) is a nonsingular Poisson variety, (1) There is a natural identification EH ( Y, Λ ) X ( k [ ǫ ]) ∼ = H ( Y, ( ∧ T Y ⊕ i ∗ N X/Y ) • )(2) Given an infinitesimal simultaneous deformation η of X in ( Y, Λ ) over A ∈ Art and a smallextension e : 0 → ( t ) → ˜ A → A → , we can associate an element o η ( e ) ∈ H ( Y, ( ∧ T Y ⊕ i ∗ N X/Y ) • ) ,which is zero if and only if there is a lifting of η to ˜ A .Proof. Let U = { U i } be an affine open covering of Y and let I i = ( f i , ..., f Ni ) be a Poisson ideal of Γ( U i , O Y )defining U i ∩ X such that { f i , · · · f Ni } is a regular sequence. We keep the notations in subsection B.2.A first-order simultaneous deformation of X in ( Y, Λ ) is a flat family X −−−−→ X ⊂ ( Y × Spec ( k [ ǫ ]) , Λ + ǫ Λ ′ ) y y Spec ( k [ ǫ ]) −−−−→ Spec ( k [ ǫ ])Since Λ + ǫ Λ ′ with Λ ′ ∈ H ( Y, ∧ T Y ) define a Poisson structure on Y × Spec ( k [ ǫ ]), we have [Λ , Λ ′ ] = 0.Assume that X is determined by a Poisson ideal sheaf I generated by { f αi + ǫg αi } , α = 1 , ..., N for some g αi ∈ Γ( U i , O Y ). Then (¯ g i , ..., ¯ g Ni ) define a global section in H ( X, N X/Y ) as in the proof of PropositionB.2.4. On the other hand, ( f i + ǫg i , ..., f Ni + ǫg Ni ) is a Poisson ideal, we have [Λ + ǫ Λ ′ , f αi + ǫg αi ] = P Nβ =1 ( f βi + ǫg βi )( T βiα + ǫW βiα ) for some W βiα ∈ Γ( U i , T Y ). Then [Λ , g αi ] + [Λ ′ , f αi ] = P Nβ =1 ¯ g βi ¯ T βiα so that˜ ∇ ( − Λ ′ , (¯ g i , ..., ¯ g Ni )) = 0. Hence ( − Λ ′ , { (¯ g i , ..., ¯ g Ni ) } ) ∈ H ( Y, ( ∧ T Y ⊕ i ∗ N X/Y ) • ).Next we identify obstructions. Consider a small extension e : 0 → ( t ) → ˜ A → A →
0. Let η :=( X ⊂ ( Y × Spec ( A ) , Λ)) be an infinitesimal simultaneous deformation of X in ( Y, Λ ) over A . Then X isdetermined by a Poisson ideal sheaf I A generated by ( F i , ..., F Ni ) in Γ( U i , O Y ) ⊗ A such that F αi ≡ f αi ⊗ m A , and { F i , ..., F Ni } is a regular sequence. Let Λ i ∈ Γ( U i , ∧ T Y ) ⊗ A be the restriction of Λ on U i . Since ( F i , ..., F Ni ) = ( F j , .., F Nj ), we have F αi = P Nβ =1 R αijβ F βj for some R αijβ ∈ Γ( U i ∩ U j , O Y ) ⊗ A .Since ( F i , ..., F Ni ) is a Poisson ideal, we have [Λ i , F αi ] = P Nβ =1 F βi W βiα for some W βiα ∈ Γ( U i , T Y ) ⊗ A . Let˜Λ i ∈ Γ( U i , ∧ T Y ) ⊗ ˜ A be an arbitrary lifting of Λ i , ˜ F αi ∈ Γ( U i , O Y ) ⊗ ˜ A be an arbitrary lifting of F αi ,˜ T βiα ∈ Γ( U i , T Y ) ⊗ ˜ A be an arbitrary lifting of W βiα , and ˜ R αijβ ∈ Γ( U i , O Y ) ⊗ ˜ A be an arbitrary lifting of R αijβ . Then [˜Λ i , ˜Λ i ] = t Π i for some Π i ∈ Γ( U i , ∧ T Y ), ˜Λ i − ˜Λ j = t Λ ′ ij for some Λ ′ ij ∈ Γ( U i ∩ U j , ∧ T Y ),[˜Λ i , ˜ F αi ] − P Nβ =1 ˜ F βi ˜ T βiα = tG αi for some G αi ∈ Γ( U i , T Y ), ˜ F αi − P Nβ =1 ˜ R αijβ ˜ F βj = th αij for some h αij ∈ Γ( U i ∩ U j , O Y ), and ˜ R αikγ − P Nβ =1 ˜ R αijβ ˜ R βjkγ = tP αijkγ for some P αijkγ ∈ Γ( U i ∩ U j ∩ U k , O Y ). We will show that( { Π i } , { ( − ¯ G i , ..., − ¯ G Ni ) } ) ⊕ ( {− Λ ′ ij } , { ¯ h ij , ..., ¯ h Nij } ) ∈ C ( U , ∧ T Y ⊕ i ∗ ( N X/Y ⊗ T Y | X )) ⊕ C ( U , ∧ T Y ⊕ i ∗ N X/Y ) define a 1-cocycle in the following ˇCech resolution of ( ∧ T Y ⊕ i ∗ N X/Y ) • : C ( U , ∧ T Y ⊕ i ∗ ( N X/Y ⊗ ∧ T Y | X )) ˜ ∇ x C ( U , ∧ T Y ⊕ i ∗ ( N X/Y ⊗ T Y | X )) δ −−−−→ C ( U , ∧ T Y ⊕ i ∗ ( N X/Y ⊗ T Y | X )) ˜ ∇ x ˜ ∇ x C ( U , ∧ T Y ⊕ i ∗ N X/Y ) − δ −−−−→ C ( U , ∧ T Y ⊕ i ∗ N X/Y ) δ −−−−→ C ( U , ∧ T Y ⊕ i ∗ N X/Y )First we show that ˜ ∇ (( { Π i } , { ( − ¯ G i , ..., − ¯ G Ni ) } )) = 0. From (A.0.16), we have − [ 12 Π i , Λ ] = 0(C.2.3)On the other hand, as in (B.2.2), we can show P Nγ =1 F γi ([Λ i , W γiα ] − P Nβ =1 W γiβ ∧ W βiα ) so that P Nγ =1 ˜ F γi ([Λ i , ˜ T γiα ] − P Nβ =1 ˜ T γiβ ∧ ˜ T βiα ) = P Nγ =1 ˜ F γi tQ γiα = P Nγ =1 f γi tQ γiα for some Q γiα ∈ Γ( U i , ∧ T Y ). Then we have t [Λ , G αi ] = [˜Λ i , [˜Λ i , ˜ F αi ]] − N X β =1 [˜Λ i , ˜ F βi ˜ T βiα ] = t [ 12 Π i , f αi ] − N X β =1 ˜ F βi [˜Λ i , ˜ T βiα ] + N X β =1 [˜Λ i , ˜ F βi ] ∧ ˜ T βiα (C.2.4) = t [ 12 Π i , f αi ] − N X γ =1 ˜ F γi [˜Λ i , ˜ T γiα ] + N X β =1 tG βi ∧ T βiα + N X β,γ =1 ˜ F γi ˜ T γiβ ∧ ˜ T βiα By taking − on (C.2.4), we obtain t [Λ , G αi ] = t [ 12 Π i , f αi ] − N X γ =1 ¯ f γi tQ γiα + N X β =1 t ¯ G βi ∧ ¯ T βiα = t [ 12 Π i , f αi ] + N X β =1 t ¯ G βi ∧ ¯ T βiα ⇐⇒ [ 12 Π i , f αi ] − [ − ¯ G αi , Λ ] + ( − N X β =1 − ¯ G βi ∧ ¯ T βiα = 0(C.2.5)Next we show that δ (( { Π i } , { ( − ¯ G i , ..., − ¯ G Ni ) } )) + ˜ ∇ (( {− Λ ′ ij } , { ¯ h ij , ..., ¯ h Nij } )) = 0. From (A.0.17), wehave δ ( 12 Π i ) − [ − Λ ′ ij , Λ ] = 0(C.2.6)On the other hand, we have t ( G αi − N X β =1 r αijβ G βj ) = [˜Λ i , ˜ F αi ] − N X β =1 ˜ F βi ˜ T βiα − N X β =1 ˜ R αijβ [˜Λ j , ˜ F βj ] + N X β,γ =1 ˜ R αijβ ˜ F γj ˜ T γjβ (C.2.7) t [Λ , h αij ] − t N X β =1 h βij T βiα = [˜Λ i , ˜ F αi ] − N X β =1 [˜Λ i , ˜ R αijβ ˜ F βj ] − N X β =1 ˜ F βi ˜ T βiα + N X β,γ =1 ˜ R βijγ ˜ F γj ˜ T βiα (C.2.8) = [˜Λ i , ˜ F αi ] − N X β =1 ˜ F βj [˜Λ i , ˜ R αijβ ] − N X β =1 ˜ R αijβ [˜Λ j + t Λ ′ ij , ˜ F βj ] − N X β =1 ˜ F βi ˜ T βiα + N X β,γ =1 ˜ R βijγ ˜ F γj ˜ T βiα As in (B.2.1), we can show P Nγ =1 F γj ( P Nβ =1 R βijγ W βiα − [Λ i , R αijγ ] − P Nβ =1 R αijβ W γjβ ) = 0 so that we have P Nγ =1 ˜ F γj ( P Nβ =1 ˜ R βijγ ˜ T βiα − [˜Λ i , ˜ R αijγ ] − P Nβ =1 ˜ R αijβ ˜ T γjβ ) = P Nγ =1 ˜ F γj tS αijγ for some S αijγ ∈ Γ( U i ∩ U j , T Y ). Then EFORMATIONS OF COMPACT HOLOMORPHIC POISSON SUBMANIFOLDS 59 from (C.2.7) and (C.2.8), we get t ( G αi − N X β =1 r αijβ G βj ) − t [Λ , h αij ] + t N X β =1 h βij T βiα (C.2.9) = N X β,γ =1 ˜ R αijβ ˜ F γj ˜ T γjβ + N X γ =1 ˜ F γj [˜Λ i , ˜ R αijγ ] + N X β =1 r αijβ [ t Λ ′ ij , f βj ] − N X β,γ =1 ˜ R βijγ ˜ F γj ˜ T βiα By taking − on (C.2.9), we get t ( ¯ G αi − N X β =1 ¯ r αijβ ¯ G βj ) − t [Λ , h αij ] + t N X β =1 ¯ h βij ¯ T βiα = − N X γ =1 t ¯ S ijγ ˜ f γi + N X β =1 t ¯ r αijβ [Λ ′ ij , f βj ] = N X β =1 t ¯ r αijβ [Λ ′ ij , f βj ] = t [Λ ′ ij , f βi ] ⇐⇒ ( N X β =1 ¯ r αijβ · ( − ¯ G βj ) − ( − ¯ G αi )) + [ − Λ ′ ij , f βi ] − [ h αij , Λ ] + N X β =1 ¯ h βij ¯ T βiα = 0(C.2.10)Lastly, from (A.0.18) and (B.2.12), we have δ ( {− Λ ′ ij } , { (¯ h ij , ..., ¯ h Nij ) } ) = 0 . (C.2.11)Hence from (C.2.3),(C.2.5),(C.2.6), (C.2.10), and (C.2.11), ( { Π i } , { ( − ¯ G i , ..., − ¯ G Ni ) } ) ⊕ ( {− Λ ′ ij } , { (¯ h ij , ..., ¯ h Nij ) } ) ∈ C ( U , ∧ T Y ⊕ i ∗ ( N X/Y ⊗ T Y | X )) ⊕ C ( U , ∧ T Y ⊕ i ∗ N X/Y ) define a 1-cocycle in the above ˇCech resolution.Now we choose another arbitrary lifting F ′ αi ∈ Γ( U i , O Y ) ⊗ ˜ A of F αi , another arbitrary lifting ˜ T ′ βiα ∈ Γ( U i , T Y ) ⊗ ˜ A of W βiα , another arbitrary lifting R ′ αijβ ∈ Γ( U i ∩ U j , O Y ) ⊗ ˜ A of R αijβ and another arbitrary lifting˜Λ ′ i ∈ Γ( U i , ∧ T Y ) ⊗ ˜ A of Λ i . We show that the associated cohomology class b := ( { Π ′ i } , { ( − ¯ G ′ i , ..., − ¯ G ′ Ni ) } ) ⊕ ( {− Λ ′′ ij } , { (¯ h ′ ij , ..., ¯ h ′ Nij ) } ) is cohomologous to a := ( { Π i } , { ( − ¯ G i , ..., − ¯ G Ni ) } ) ⊕ ( {− Λ ′ ij } , { (¯ h ij , ..., ¯ h Nij ) } ).We note that ˜ F ′ αi = ˜ F αi + tA αi for some A αi ∈ Γ( U i , O Y ), ˜ T ′ βiα = ˜ T βiα + tB βiα for some B βiα ∈ Γ( U i , T Y ),˜ R ′ αijβ = ˜ R αijβ + tC αijβ for some C αijβ ∈ Γ( U i ∩ U j , O Y ) and ˜Λ ′ i = ˜Λ i + tD i for some D i ∈ Γ( U i , ∧ T Y ). Then t ( G ′ αi − G αi ) = [˜Λ ′ i , ˜ F ′ αi ] − N X β =1 ˜ F ′ βi ˜ T ′ βiα − [˜Λ i , ˜ F αi ] + N X β =1 ˜ F βi ˜ T βiα = [ tD i , f αi ] + [Λ , tA αi ] − N X β =1 tA βi T βiα − N X β =1 f βi tB βiα (C.2.12)By taking − on (C.2.12), we get¯ G ′ αi − ¯ G αi = [ D i , f αi ] + [Λ , ¯ A αi ] − N X β =1 ¯ A βi ¯ T βiα ⇐⇒ − ¯ G αi − ( − ¯ G ′ αi ) = [ D i , f αi ] − [Λ , − A αi ] + N X β =1 − ¯ A βi ¯ T βiα (C.2.13)and from (B.2.14), we have ¯ h ′ αij − ¯ h αij = ¯ A αi − P Nβ =1 ¯ r αijβ ¯ A βj so that { ¯ h αij − ¯ h ′ αij } = − δ ( {− ¯ A αi } ). On theother hand, from (A.0.19) and (A.0.20), we have Π i − Π ′ i = − [ D i , Λ ] and − Λ ′ ij − ( − Λ ′′ ij ) = − δ ( D i ).Hence ( { D i } ⊕ { ( − ¯ A i , ..., − ¯ A Ni ) } ) is mapped to a − b so that a is cohomologous to b . So given a smallextension e : 0 → ( t ) → ˜ A → A →
0, we can associate an element o η ( e ) := the cohomology class a ∈ H ( Y, ( ∧ T Y ⊕ i ∗ N X/Y ) • ). We note that o η ( e ) = 0 if and only if there exists collections { ˜ F αi } , { ˜ T βiα } , { ˜ R αijβ } and { ˜Λ i } such that ¯ h αij = 0 , ¯ G αi = 0 , α = 1 , ..., N, Π i = 0, and Λ ′ ij = 0:(1) If ¯ h αij = 0, then ( ˜ F i , ..., ˜ F Ni ) = ( ˜ F j , ..., ˜ F Nj ) so that { ( ˜ F i , ..., ˜ F iN ) } define an ideal sheaf on Y × Spec ( ˜ A ).(2) If Π i = 0 and Λ ′ ij = 0, then [˜Λ i , ˜Λ i ] = 0, and { ˜Λ i } glues together to define a Poisson structure on Y × Spec ( ˜ A ).(3) If ¯ G αi = 0, then G αi = P Nβ =1 f βi P βi for some P βi ∈ Γ( U i , T Y ). Then [˜Λ i , ˜ F αi ] = P Nβ =1 ( ˜ T βi + tP βi ) ˜ F βi so that ( ˜ F i , ... ˜ F Ni ) defines a Poisson ideal.Hence o η ( e ) = 0 if and only if there is a lifting of η to ˜ A . (cid:3) References [BHPVdV04] Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven,
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