Deformations of Generalized Kahler Structures and Bihermitian Structures
aa r X i v : . [ m a t h . DG ] O c t Deformations of Generalized K¨ahler Structures andBihermitian Structures
Ryushi Goto ∗ Abstract
Let (
X, J ) be a compact K¨ahler manifold with a non-zero holomorphic Poisson struc-ture β . If the obstruction space for deformations of generalized complex structureson ( X, J ) vanishes, we obtain a family of deformations of non-trivial bihermitianstructures (
J, J − t , h t ) on X by using β . In addition, if the class [ β · ω ] does notvanish for a K¨ahler form ω , then the complex structure J − t is not equivalent to J for small t = 0 under diffeomorphisms. Our method is based on the construction ofgeneralized complex and generalized K¨ahler structures developed in [10] and [11].As applications, we obtain such deformations of bihermitian structures on del Pezzosurfaces, the Hirtzebruch surfaces F , F and degenerate del Pezzo surfaces. Furtherwe show that del Pezzo surfaces S n (5 ≤ n ≤ F and degenerate del Pezzo sur-faces admit bihermitian structures for which ( X, J − t ) is not biholomorphic to ( X, J )for small t = 0. Contents
Mathematics Subject Classification . Primary 53C25; Secondary 53C55.
Key words and phrases . bihermitian structures, generalized complex , generalized K¨ahler structures,deformation theory, Maurer-Cartan equation ∗ Partly supported by the Grant-in-Aid for Scientific Research (C), Japan Society for the Promotionof Science. Construction of deformations of bihermitian structures with J + t = J
185 The convergence 226 Applications 23 F and F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.5 Bihermitian structures on degenerate del Pezzo surfaces . . . . . . . . . . . 30 Introduction
A bihermitian structure on a C ∞ manifold X consists of a pair of integrable complexstructures J + and J − with a Riemannian metric h which is hermitian with respect toboth J + and J − . If a complex manifold ( X, J ) has a bihermitian structure ( J + , J − , h )with the property J + = J , then we say that ( X, J ) admits a (compatible) bihermitianstructure. A bihermitian structure ( J + , J − , h ) is distinct if the complex manifold ( X, J + )is not biholomorphic to ( X, J − ). We have the two ∂ -operators ∂ + and ∂ − correspondingto the complex structures J + and J − respectively. In this paper we always assume thata bihermitian structure satisfies the condition, − d c + ω + = d c − ω − = db, (0.1)where d c ± = √− ∂ ± − ∂ ± ) and ω ± denote the fundamental 2-forms with respect to J ± and b is a real 2-form. (Note that if H := − d c + ω + = d c − ω − is not d -exact but d -closed,( J + , J − , h ) is called the H -twisted bihermitian structure.) There is a research of compactcomplex surfaces which admit bihermitian structures from the view point of Riemanniangeometry [2]. Bihermitian structures with the condition (0.1) appeared on the target spaceof (2 ,
2) supersymmetric sigma model [7]. Surprisingly it turned out that there is a oneto one correspondence between generalized K¨ahler structures and bihermitian structureswith the condition (0.1)[12]. It is thus expected that the construction of interesting andvarious generalized K¨ahler structures would be a major step of development of the theoryof bihermitian structures. Let (
X, J ) be a compact K¨ahler manifold with a K¨ahler form2 . In the paper [10, 11], the author constructed a family of deformations of bihermitianstructures by using a holomorphic Poisson structure β . In the present paper, we shallobtain another family of deformations of bihermitian structures ( J + t , J − t , h t ) of ( X, J ),starting with the ordinary K¨ahler structure which satisfies J + t = J for all t , J − = J and J − t = ± J for small t = 0, where t is a parameter of deformations.Throughout this paper we will assume that X is the underlying differential manifoldof a complex manifold M = ( X, J ) with the structure sheaf O M . We denote by Θ thesheaf of germs of sections of the tangent bundle T , J of M = ( X, J ) and ∧ p Θ is the sheafof germs of p -th skew symmetric tensors of Θ. Our main theorem is the following : Theorem . Let M = ( X, J ) be a compact K¨ahler manifold. We assume thatthe direct sum of cohomology groups ⊕ i =0 H i ( M, ∧ − i Θ) vanishes. Then for every K¨ahlerform ω and every non-zero holomorphic Poisson structure β , there exist deformations ofbihermitian structures ( J + t , J − t , h t ) which satisfies, J + t = J − = J, ddt J − t | t =0 = − β · ω + β · ω ) , (0.2) where β · ω is the ∂ -closed forms of type (0 , with coefficients in the tangent bundle T , J which is given by the contraction between β and ω , and β · ω is the complex conjugate.The ∂ -closed form β · ω gives rise to the Kodaira-Spencer class − β · ω ] ∈ H ( M, Θ) ofdeformations { J − t } . The condition (0.2) implies that J − t = ± J for small t = 0 as almost complex structures.However these J − t and J might be equivalent under diffeomorphisms. If the class [ β · ω ] ∈ H ( M, Θ) does not vanish, the family of deformations { J − t } is not obtained by the actionof a one-parameter family of diffeomorphisms on J . Thus the complex manifold ( X, J − t )is different from ( X, J ) for small t = 0. Theorem . Let M = ( X, J ) be a compact K¨ahler manifold. We assume that thedirect sum of cohomology groups ⊕ i =0 H i ( M, ∧ − i Θ) vanishes and in addition, the class [ β · ω ] ∈ H ( M, Θ) does not vanish for a K¨ahler form ω and a holomorphic Poissonstructure β . Then there exist deformations of distinct bihermitian structures ( J, J − t , h t ) ,that is, ( X, J − t ) is not biholomorphic to M = ( X, J ) for small t = 0 . The infinitesimal deformations of generalized complex structures are given by thedirect sum of cohomology group, H ( M, ∧ Θ) ⊕ H ( M, Θ) ⊕ H ( M, O X ) , where H ( M, Θ) is the space of the Kodaira-Spencer classes which gives infinitesimaldeformations of (usual) complex structures. The cohomology group H ( M, O X ) corre-sponds to the exponential action of ∂ -closed 2-form of type (0 , b fields . The space H ( M, ∧ Θ) corresponds to deformations of gen-eralized complex structures {J βt } by a Poisson structure β which are called the Poissondeformations . As in deformations of complex manifolds, there exits an obstruction todeformations of generalized complex structures in general. The obstruction space to de-formations of generalized complex structures at J J is given by the direct sum of theordinary cohomology groups, M i =0 H i ( M, ∧ − i Θ) := H ( M, ∧ Θ) ⊕ H ( M, ∧ Θ) ⊕ H ( M, Θ) ⊕ H ( M, O M )which is the obstruction space in the theorem 0.1. If the space of the obstruction vanishes,we can apply the method in [10] and [11] to construct a family of generalized K¨ahlerstructures which corresponds to the one of bihermitian structures in the theorem 0.1.More precisely, the complex structure J gives a generalized complex structure J J andthe K¨ahler structure ω also provides the d -closed non-degenerate, pure spinor ψ = e √− ω which induces the generalized complex structure J ψ . The pair ( J J , ψ ) gives rise to ageneralized K¨ahler structure ( J J , J ψ ). It is the essential feature that the generalizedgeometry inherits the symmetry of the Clifford group of the direct sum of the tangentbundle T and the cotangent bundle T ∗ on a manifold X . The space of almost generalizedK¨ahler structures forms an orbit by the diagonal action of the Clifford group. Thuswe construct deformations of almost generalized K¨ahler structures with one pure spinorstaring with ( J J , ψ ) by the action of the Clifford group, J t = Ad e Z ( t ) J J , ψ t = e Z ( t ) ψ, where e Z ( t ) is a family of the Clifford group and Ad e Z ( t ) denotes the adjoint action of theClifford group on J J (see [10]). The pair ( J t , ψ t ) induces the almost generalized K¨ahlerstructure ( J t , J ψ t ) and then the corresponding bihermitian structures ( J + t , J − t , h t ) aregiven by the action of Γ ± t ∈ GL(
T X ) by J ± t = (Γ ± t ) − ◦ J ◦ Γ ± t , where Γ ± t is explicitlydescribed in terms of Z ( t ) (see 3.4 in section 3). Thus our problem is reduced to construct Z ( t ) which satisfies the following three conditions: J t := Ad e Z ( t ) J J are integrable generalized complex structures (0.3) dψ t := de Z ( t ) ψ = 0 (0.4)(Γ + t ) − ◦ J ◦ Γ + t = J (0.5)Then Z ( t ) yields deformations ( J t , J ψ t ) of generalized K¨ahler structures which gives riseto bihermitian structures in the theorem 0.1 (see section 4 for more detail). Note that if ψ t is closed, the induced structure J ψ t is integrable. J, J − t , h t )by the Hamiltonian diffeomorphisms on del Pezzo surfaces and Gualtieri [13] extendedthe approach to higher dimensional Poisson manifolds. The bihermitian structures whichthey constructed give the equivalent two complex structures under diffeomorphisms. Ourconstructions enable us to obtain distinct bihermitian structures.In section 1, we will give a short explanation of deformations of generalized complexstructures. Deformations of generalized complex structures are often described in thelanguage of complex Lie algebroid [20], [12]. It is necessary to translate it in the terms ofthe action of the (real) Clifford group for our construction of generalized K¨ahler structures.In section 2, we recall the stability theorem of generalized K¨ahler structure with one purespinor which was shown in [10] and [11]. In section 3, we give a description of Γ ± t whichgives deformations of bihermitian structures corresponding to the ones of generalizedK¨ahler structures. In section 4, we will construct deformations of bihermitian structuresin the main theorem 0.1 as formal power series. In section 5, we will show the convergenceof the power series constructed in section 4 and finish our proof of the main theorem. Insection 6, we apply our method to complex surfaces. In the case of complex surfaces,we only need to show that the cohomology groups H ( M, K − M ) and H ( M, Θ) vanish toobtain deformations in the theorem 0.1, where K M is the canonical line bundle.In subsection 6.1, we show that every del Pezzo surface admits deformations of biher-mitian structures as in theorem 0.1. Let S n be a del Pezzo surface which is the blow-upof C P at n points. Then we prove that if n ≥
5, there exists a class [ β · ω ] ∈ H ( S n , Θ)which does not vanish for a K¨ahler form ω . As a result, we obtain distinct bihermitianstructures on S n ( n ≥ H ( M, K − M ) and H ( M, Θ) on a complex surface M . In subsection 6.3 we will show thenon-vanishing theorem of the class [ β · ω ] ∈ H ( M, Θ) which gives rise to unobstructed de-formations. Applying these vanishing theorems and the non-vanishing theorem, we obtainbihermitian structure ( J + , J − ) on F = ( X, J ) on which the complex manifold (
X, J + )is F and ( X, J − ) is C P × C P in subsection 6.4. We also show that the Hirtzebruchsurface F admits bihermitian structures. Degenerate del Pezzo surfaces are the blow-upof C P at r points, 0 ≤ r ≤ almost general position (see subsection 6.5for more detail). It turns out that the obstruction spaces still vanish on degenerate delPezzo surfaces. If the anti-canonical line bundle is not ample, then there is a ( − C with K · C = 0 and it follows that the class [ β · ω ] does not vanish. Hence we obtainbihermitian structures in theorem 0.1 on the degenerate del Pezzo surfaces which yielddistinct two complex manifolds. We contract all ( − The author received a note that Gualtieri also developed a modified approach to obtain bihermitianstructures on F recently.
5o obtain a del Pezzo surface with rational double points, which is called the
Gorensteinlog del Pezzo surface , [5, 4, 1]. In appendix I, we give the power series construction of theKuranishi family of generalized complex structures. In appendix II, we collect necessaryformulae and give an explanation of the Schouten bracket and the Jacobi identity of thebrackets.The author would like to thank for Professor A. Fujiki for valuable comments. He isalso thankful to Professor V. Apostolov and Professor M. Gualtieri for our remarkablediscussion at Montreal.
Let J be a generalized complex structure on a compact manifold X of real dimension 2 n .Then the generalized complex structure J gives the decomposition ( T ⊕ T ∗ ) C = L J ⊕ L J ,where L J denotes the eigenspace with eigenvalue √− L J is the complex conjugateof L J . For a section ε ∈ ∧ L J , the exponential e ε is regarded as the section of complexClifford group which also induces the section of SO( T ⊕ T ∗ , C ) by the adjoint action Ad e ε .Small deformations of almost generalized complex structure J are written by a sectionof ∧ L J , J t = Ad e ε ( t ) J, where t is the parameter of deformations. The integrability condition of almost generalizedcomplex structure J t is given by the Maurer-Cartan equation, d L ε ( t ) + 12 [ ε ( t ) , ε ( t )] S = 0 , (1.1)where d L : ∧ k L J → ∧ k +1 L J denotes the exterior derivative of the Lie algebroid L J and the bracket [ , ] S is the Schouten bracket of L J . Hence the problem of deformationsreduces to solving the Maurer-Cartan equation,At first, we write a family of sections ε ( t ) as a power series in tε ( t ) = ε t + ε t
2! + · · · , (1.2)Note that our power series starts from ε . Substituting the power series (1.2) into theMaurer-Cartan equation, we obtain the equation on t . We denote by ([ ε ( t ) , ε ( t )] S ) [ k ] the k th homogeneous term in t . Then the equation is reduced to infinitely many equations,1 k ! d L ε k + 12 ([ ε ( t ) , ε ( t )] S ) [ k ] = 0 (1.3)We have the differential complex ( ∧ • L J , d L ) which is elliptic,0 → L J d L → ∧ L J d L → ∧ L J d L → · · · k = 1, the equation is d L ε = 0. It implies that ε is a section of ∧ L J which is d L -closed. We take ε as a Hormonic section which satisfies △ L ε = ( d L d ∗ L + d ∗ L d L ) ε = 0 , where d ∗ L is the formal adjoint operator of d L with respect to a Riemannian metric on M .There are actions of diffeomorphisms and d -exact b -fields on generalized complex struc-tures which generate d L -exact sections of ∧ L J infinitesimally. We identify deformationsby both actions of diffeomorphisms and d -exact b -fields. It implies that the infinitesimaldeformations (the first order deformations) are given by the cohomology group H ( L J )of the elliptic differential complex ( ∧ • L J , d L ) . The third cohomology group H ( L J )is regarded as the space of the obstructions to deformations. The deformation theoryof generalized complex structures was already discussed in [12] by the implicit functiontheorem. We will give the different construction of deformations of generalized complexstructures by using the power series, which is analogous to the one of original Kodaira-Spencer theory. Our method yields an estimate of the convergent series which is necessaryfor the construction of generalized K¨ahler and bihermitian structures. Theorem . If the cohomology group H ( L J ) vanishes, then we have a family ofdeformations of generalized complex structures which are parametrized by an open set of H ( L J ) . Proof.
We solve the equations (1.3) by the induction on the degree of t . We assumethat there are sections ε , · · · , ε k − ∈ ∧ L J which satisfy the equations1 i ! ε i + 12 ([ ε ( t ) , ε ( t )] S ) [ i ] = 0 , (1.4)for all i < k .Then we shall show that there is a section ε k which satisfies the equation (1.3). TheSchouten bracket and the Lie algebroid derivative d L satisfy the following relations forsections ε , ε ∈ ∧ • L J , Proposition . [ ε , ε ] S = ( − | ε | | ε | [ ε , ε ] d L [ ε , ε ] S = [ d L ε , ε ] S + ( − | ε | [ ε , d L ε ] S ( − | ε | | ε | (cid:2) [ ε , ε ] , ε (cid:3) + ( − | ε | | ε | (cid:2) [ ε , ε ] , ε (cid:3) + ( − | ε | | ε | (cid:2) [ ε , ε ] , ε (cid:3) = 0 where we denote by | ε i | the degree of ε i . ε i is even and we have the ordinary Jacobi identity. The k th order term of theSchouten bracket [ ε ( t ) , ε ( t )] S is given by([ ε ( t ) , ε ( t )] S ) [ k ] = X i + j = k n + 1. We put k P ( t ) k s = P k k P k k s t k . Given two power series P ( t ) , Q ( t ), if k P k k ≤ k Q k k for all k , then we denote it by P ( t ) << Q ( t ) . For a positive integer k , if k P i k ≤ k Q i k for all i ≤ k , we write it by P ( t ) << k Q ( t ) .
8e also use the following notation. If P i = Q i for all i ≤ k , we write it by P ( t ) ≡ k Q ( t ) . (1.10)Let M ( t ) be a convergent power series defined by M ( t ) = ∞ X ν =1 c ( ct ) ν ν = ∞ X ν =1 M ν t ν , (1.11)for a positive constant c , which is determined later suitably. The key point is the followinginequality, M ( t ) << c M ( t ) (1.12)We put λ = c − . Then we also have e M ( t ) << λ e λ M ( t ) . (1.13)We assume that our power series ε ( t ) satisfies the inequality for an integer k > k ε ( t ) k s << k − M ( t ) (1.14)We apply the standard estimate of elliptic differential operators to obtain an estimate ofthe solution ε [ k ] in (1.9),2 k ! k ε k k s ≤ C k (cid:0)(cid:2) ε ( t ) , ε ( t ) (cid:3) S (cid:1) [ k ] k s − = X i + j = k
1, then it follows that k ! k ε k k s < M k . Itimplies that ε ( t ) is a convergent series. From our construction, the series ε ( t ) satisfies ε ( t ) = ε t − d ∗ L G L [ ε ( t ) , ε ( t )] S (1.18)Since H ( ∧ • L J ) = { } , we have a differential equation △ L ε ( t ) + 12 d ∗ L [ ε ( t ) , ε ( t )] S = 0 , which is elliptic for sufficiently small ε ( t ). Thus it follows that ε ( t ) is a smooth solution.9n the appendix, we further construct the Kuranishi family of deformations of gener-alized complex structures which gives the space of deformations even in the cases wherethe obstruction space H ( ∧ • L J ) does not vanish.We denote by CL the Clifford algebra bundle of T ⊕ T ∗ on a manifold X which admitsfiltrations of even degree and odd degree,CL ⊂ CL ⊂ CL ⊂ · · · CL ⊂ CL ⊂ CL ⊂ · · · , where CL = T ⊕ T ∗ and CL denotes the the subbundle of CL which consists of elementsof degree 2 or 0, (for simplicity, we call CL the Clifford algebra of T ⊕ T ∗ .) Let J be ageneralized complex structure on X which gives the decomposition,( T ⊕ T ∗ ) C = L J ⊕ L J We denote by ∧ p L J the bundle of the p th skew symmetric tensor of L J . Let U − n J be the line bundle of ( X, J ) which consists of non-degenerate, complex pure spinorscorresponding to J . We call U − n J the canonical line bundle K J of J . There is the actionof T ⊕ T ∗ on differential forms ∧ • T ∗ by the interior product and the exterior productwhich induces the spin representation of the Clifford algebra CL on ∧ • T ∗ . By the actionof ∧ p L J on K J , we have the vector bundles, U − n + p := ∧ p L J · K J Then the differential forms ∧ • T ∗ on X are decomposed into ∧ • T ∗ = n M p =0 U − n + p We denote by π U − n + p the projection to the bundle U − n + p . The set of almost generalizedcomplex structures forms the orbit of the (real) Clifford group of the Clifford algebra CLof T ⊕ T ∗ which acts on J by the adjoint action. The Lie algebra of the Clifford groupis the subalgebra CL . Small deformations of almost generalized complex structures {J t } are given in terms of the adjoint action, J t := Ad e a ( t ) J , where a ( t ) = a t + a t + · · · is a CL -valued power series in t .In order to obtain deformations of generalized K¨ahler structures, we need to considera section a ( t ) of the bundle CL ( T ⊕ T ∗ ).it is crucial that the set of almost generalized K¨ahler structures just forms an orbitof the action of the real Clifford group and deformations of almost generalized K¨ahler10tructures are not given by the action of the complex Clifford group. The followinglemma is necessary for the construction of generalized K¨ahler structures. which is alreadyproved in [10]. Lemma . For small deformations of almost generalized complex structures givenby J t := Ad e ε ( t ) J as before, there exists a unique family of sections a ( t ) of real Cliffordbundle CL such that J t = Ad e a ( t ) J , and a ( t ) is in the real part of ∧ L J ⊕ ∧ L J . Conversely, If we have a family of de-formations of almost generalized complex structure J t = Ad e a ( t ) J which is given by theaction of a section a ( t ) ∈ CL , then there exists a unique section ε ( t ) ∈ ∧ L J such that J t = Ad e ε ( t ) J . We consider the operator e − a ( t ) ◦ d ◦ e a ( t ) acting on K J = U − n . Then as discussed in[9], the operator e − a ( t ) ◦ d ◦ e a ( t ) is a Clifford-Lie operator of order 3 whose image is in U − n +1 ⊕ U − n +3 .It is shown in [10] that the almost generalized complex structure J t = Ad e a ( t ) J isintegrable if and only if the projection to the component U − n +3 vanishes, that is, π U − n +3 e − a ( t ) ◦ d ◦ e a ( t ) = 0We denote by (cid:0) π U − n +3 e − a ( t ) ◦ d ◦ e a ( t ) (cid:1) [ k ] the k th term of π U − n +3 e − a ( t ) ◦ d ◦ e a ( t ) . Let J J be the generalized complex structure on X defined by a ordinary complex structure J .We put M = ( X, J ). Then as in [10] and [11], the obstruction space to deformations of J J is given by ⊕ p + q =3 H p ( M, ∧ q Θ). In this case, the canonical line bundle is the ordinaryone K J which consists of complex forms of type ( n, Proposition . Let M = ( X, J ) be a compact K¨ahler manifold with a K¨ahler form ω . We assume that the cohomology groups ⊕ p + q =3 H p ( M, ∧ q Θ) vanish. If there is a set ofsections a , · · · a k − of CL which satisfies (cid:0) π U − n +3 e − a ( t ) de a ( t ) (cid:1) [ i ] = 0 , for all i < k, (1.19) and k a ( t ) k s << k − C M ( t ) , then there is a section a k of CL which satisfies the followings: π U − n +3 (cid:0) e − a ( t ) de a ( t ) (cid:1) [ k ] = 0 and k a ( t ) k s << k C M ( t ) , where a ( t ) = P ∞ i =1 1 i ! a i t i and M ( t ) is the convergent series in(1.12) and C is a positive constant. roof. We use the notation as in (1.10). The equation (1.19) is equivalent to saythat there is a section ˆ E ( t ) ∈ T ⊕ T ∗ such that , e − a ( t ) de a ( t ) · φ ≡ k − ˆ E ( t ) · φ, (1.20)for all φ ∈ K J . By the left action of e a ( t ) on both sides of the equation (1.20), we have de a ( t ) · φ ≡ k − e a ( t ) ˆ E ( t ) · φ. We put E ( t ) = e a ( t ) ˆ E ( t ) e − a ( t ) . Then it follows that de a ( t ) · φ ≡ k − E ( t ) · e a ( t ) φ. (1.21)From the lemma 1.3, we have ε ( t ) ∈ ∧ L J such that e ε ( t ) · φ ≡ k − e a ( t ) · φ. (1.22)Substituting (1.22) into (1.21), we obtain de ε ( t ) · φ ≡ k − E ( t ) · e ε ( t ) φ. (1.23)By the right action of e − ε ( t ) on (1.23) again, we have e − ε ( t ) de ε ( t ) · φ ≡ k − e − ε ( t ) E ( t ) · e ε ( t ) φ ≡ k − ˜ E ( t ) · φ, (1.24)where ˜ E ( t ) = e − ε ( t ) E ( t ) · e ε ( t ) . Thus as in [10], the equation(1.24) is equivalent to theMaurer-Cartan equation, d L ε ( t ) + 12 [ ε ( t ) , ε ( t )] S ≡ k − . Then as is shown in the theorem 1.1, there is a section ε k such that d L ε ( t ) + 12 [ ε ( t ) , ε ( t )] S ≡ k . We define a k by a k = ε k + ε k . Then it follows that e − a ( t ) de a ( t ) · φ ≡ k ˆ E ( t ) · φ. Hence we have (cid:0) π U − n +3 e − a ( t ) de a ( t ) (cid:1) [ k ] = 0 and k a ( t ) k s << k C M ( t ).12 Deformations of generalized K¨ahler structures
Let (
X, J, ω ) be a compact K¨ahler manifold and ( J , J ψ ) the generalized K¨ahler struc-ture induced from ( J, ω ) by J = J J and ψ = e √− ω . Since two generalized complexstructures J and J ψ are commutative, the generalized K¨ahler structure ( J , J ψ ) gives thesimultaneous decomposition of ( T ⊕ T ∗ ) C ,( T ⊕ T ∗ ) C = L + J ⊕ L −J ⊕ L + J ⊕ L −J , where L + J ⊕ L −J is the eigenspace with eigenvalue √− J and L + J ⊕ L −J is the eigenspace with eigenvalue √− J ψ and L ±J denotes the complexconjugate. In [10, 11], the author showed the stability theorem of generalized K¨ahlerstructures with one pure spinor, which implies that if there is a one dimensional analyticdeformations of generalized complex structures {J t } parametrized by t , then there existsa family of non-degenerate, d -closed pure spinor ψ t such that the family of pairs ( J t , ψ t )becomes deformations of generalized K¨ahler structures starting from ( J , ψ ) = ( J , ψ ).As in section 2, small deformations J t can be written by the adjoint action of a ( t ) in CL , J t := Ad e a ( t ) J . Then we can obtain a family of real sections b ( t ) of the bundle ( L −J · L + J ⊕ L −J · L + J )such that ψ t = e a ( t ) e b ( t ) ψ is the family of non-degenerate, d -closed pure spinor ψ t . Thebundle K = U , − n +2 is generated by the action of real sections of ( L −J · L + J ⊕ L −J · L + J )on ψ (see page 125 in [11] for more detail).We define Z ( t ) by e Z ( t ) = e a ( t ) e b ( t ) . Since Ad e b ( t ) J = J , we obtain J t = Ad e a ( t ) J = Ad e a ( t ) Ad e b ( t ) J = Ad e Z ( t ) J . Then thefamily of deformations of generalized K¨ahler structures is given by the action of e Z ( t ) ,( J t , ψ t ) = (cid:0) Ad e Z ( t ) J , e Z ( t ) · ψ (cid:1) . By the similar method as in [11] together with the proposition 1.4, we obtain the followingproposition,
Proposition . Let ( X, J, ω ) be a compact K¨ahler manifold. We assume that thecohomology groups ⊕ p + q =3 H p ( X, ∧ q Θ) vanish. If there is a set of sections a , · · · a k − ofCL which satisfies π U − n +3 (cid:0) e − a ( t ) de a ( t ) (cid:1) [ i ] = 0 , for all i < k, nd k a ( t ) k s << k − K M ( t ) for a positive constant K , then there is a set of real sections b , · · · , b k of the bundle ( L −J · L + J ⊕ L −J · L + J ) which satisfies the following equations: π U − n +3 (cid:0) e − Z ( t ) de Z ( t ) (cid:1) [ k ] = 0 (2.1) (cid:0) de Z ( t ) · ψ (cid:1) [ i ] = 0 , for all i ≤ k (2.2) k a ( t ) k s << k K λM ( t ) (2.3) k b ( t ) k s << k K M ( t ) (2.4) where a k is the section constructed in the proposition 1.4 and e Z ( t ) = e a ( t ) e b ( t ) and M ( t ) is the convergent series in (1.12) and a positive constant K is determined by λ and K .The constant λ in M ( t ) will be suitably selected to show the convergence of the powerseries Z ( t ) in section 6. We use the same notation as in pervious sections. There is a one to one correspondencebetween generalized K¨ahler structures and bihermitian structures with the condition (0.1).In this section we shall give an explicit description of Γ ± t which givens rise to bihermi-tian structure ( J + t , J − t ) corresponding to deformations ( J t , ψ t ). The correspondence isdefined at each point on a manifold, that is, the correspondence between tensor fieldswhich allows us to obtain almost bihermitian structures from almost generalized K¨ahlerstructures. The non-degenerate, pure spinor ψ t induces the generalized complex structure J ψ t . Since ( J t , J ψ t ) is a generalized K¨ahler structure and J t commutes with J ψ t , we havethe simultaneous decomposition of ( T ⊕ T ∗ ) C into four eigenspaces,( T ⊕ T ∗ ) C = L + J t ⊕ L −J t ⊕ L + J t ⊕ L −J t , where each eigenspace is given by the intersection of eigenspaces of both J t and J ψ t , L −J t = L J t ∩ L ψ t , L + J t = L J t ∩ L ψ t L + J t = L J t ∩ L ψ t , L −J t = L J t ∩ L ψ t , where L J t is the eigenspace of J t with eigenvalue √− L ψ t denotes the eigenspace of J ψ t with eigenvalue √−
1. Since J t = Ad e Z ( t ) J = e Z ( t ) J e − Z ( t ) and J ψ t = Ad e Z ( t ) J ω , wehave the isomorphism between eigenspaces,Ad e Z ( t ) : L ±J → L ±J t . Let π be the projection from T ⊕ T ∗ to the tangent bundle T . We restrict the map π tothe eigenspace L ±J which yields the map π ± t : L ±J t → T C . Let T , J ± t be the complex tangent14pace of type (1 ,
0) with respect to J ± t . Then it follows that T , J ± t is given by the image of π ± t , T , J ± t = π ± t ( L ±J t )Since deformations of generalized K¨ahler structures are given by the action of e Z ( t ) , theones of bihermitian structures J ± t should be described by the action of Γ ± t ∈ GL( T ) whichis obtained from Z ( t ). We shall describe Γ ± t in terms of a ( t ) and b ( t ). A local basis of L ±J is given by { Ad e ±√− ω V i = V i ± √− ω, V i ] } ni =1 , for a local basis { V i } ni =1 of T , J , where we regard ω as an element of the Clifford algebraand then the bracket [ ω, V i ] coincides with the interior product i V i ω . It follows that theinverse map ( π ± ) − : T , J → L ±J is given by the adjoint action of e ±√− ω ,Ad e ±√− ω = ( π ± ) − . (3.1)We define a map (Γ ± t ) , : T , J → T , J ± t by the composition,(Γ ± t ) , = π ± t ◦ Ad e Z ( t ) ◦ ( π ± ) − (3.2)= π ◦ Ad e Z ( t ) ◦ Ad e ±√− ω (3.3) L ±J Ad eZ ( t ) / / π ± (cid:15) (cid:15) L ±J t π ± t (cid:15) (cid:15) T , J (Γ ± t ) , / / T , J ± t Together with the complex conjugate (Γ ± t ) , : T , J → T , J ± t , we obtain the map Γ ± t which satisfies J ± t = (Γ ± t ) − ◦ J ◦ Γ ± t .Let J ∗ be the complex structure on the cotangent space T ∗ which is given by h J ∗ η, v i = h η, J v i , where η ∈ T ∗ and v ∈ T and h , i denote the coupling between T and T ∗ . Wedefine a map ˆ J ± : T ⊕ T ∗ → T ⊕ T ∗ by ˆ J ± ( v, η ) = v ∓ J ∗ η for v ∈ T and η ∈ T ∗ . ThenΓ ± t is written as Γ ± t = π ◦ Ad e Z ( t ) ◦ ˆ J ± ◦ Ad e ω . (3.4)The k th term of Γ ± t is denoted by (Γ ± t ) [ k ] as before. Note that (Γ ± t ) [0] =id T . We also putΓ ± t ( a ( t ) , b ( t )) = Γ ± t . Lemma . The k th term (Γ ± t ) [ k ] is given by (Γ ± t ) [ k ] = 1 k ! π ◦ (ad a k + ad b k ) ◦ ˆ J ± ◦ Ad e ω + f Γ ± k ( a Applying (3.1), for v ∈ T , J , we obtainˆ J ± ◦ Ad e ω v = Ad e ±√− ω v = ( π ± ) − v ∈ L ±J . Since ad b ( L ±J ) = [ b, L ±J ] ⊂ L ∓J and π ( L ∓J ) = T , J , thus we have π (ad b ◦ ˆ J ± ◦ Ad e ω ) v ∈ T , J .It follows that [ π (ad b ◦ ˆ J ± ◦ Ad e ω ) , J ] = 0 . The tensor space T ⊗ T ∗ defines a subbundle of CL . We denote it by T · T ∗ . Anelement γ ∈ T · T ∗ gives the endmorphism ad γ by ad γ E = [ γ, E ] for E ∈ T ⊕ T ∗ , whichpreserves the cotangent bundle T ∗ . Lemma . Let γ be an element of T · T ∗ . Then we have π ◦ (ad γ ◦ ˆ J ± ◦ Ad e ω ) = ad γ ∈ End( T ) . Proof. For a tangent vector v ∈ T , we have Ad e ω v = v + [ ω, v ] = v + ad ω v . Sincethe map ad γ preserves the cotangent T ∗ , we have ad γ ◦ ˆ J ± ◦ ad ω ( v ) ∈ T ∗ for all tangent v ∈ T . Thus it follows that π (ad γ ◦ ˆ J ± ◦ ad ω ) = 0, since π is the projection to the tangent T . Thus we obtain the result. 16 emma . We assume that there is a set of sections a , · · · , a k ∈ CL and realsections b , · · · , b k ∈ ( L −J · L + J ⊕ L −J · L + J ) which satisfies the following equations, π U − n +3 (cid:0) e − Z ( t ) d e Z ( t ) (cid:1) [ i ] = 0 , ≤ ∀ i ≤ k (cid:0) de Z ( t ) · ψ (cid:1) [ i ] = 0 , ≤ ∀ i ≤ k [(Γ ± t ) [ i ] , J ] = 0 , ≤ ∀ i < k Then the k -th term (Γ ± t ) [ k ] satisfies π U − n +3 [ d, (Γ ± t ) [ k ] ] = 0 , where [ d, (Γ ± t ) [ k ] ] is an operator from U − n = K J to U − n +1 ⊕ U − n +3 and π U − n +3 denotesthe projection to the component U − n +3 . Proof. Since we assume that the space of the obstructions to deformations of gen-eralized complex structures vanishes, we obtain a family of section ˇ a ( t ) with ˇ a i = a i for i = 1 , · · · k such that ˇ a ( t ) gives deformations of generalized complex structures, that is, π U − n +3 e − ˇ a ( t ) de ˇ a ( t ) = 0 . The the stability theorem of generalized K¨ahler structures in [10] provides deformationsof generalized K¨ahler structures with one pure spinor, (Ad e ˇ Z ( t ) J , e ˇ Z ( t ) ψ ), where e ˇ Z ( t ) = e ˇ a ( t ) e ˇ b ( t ) , where ˇ b ( t ) is a family of real sections with ˇ b i = b i , for i = 1 , · · · k . Fromthe correspondence between generalized K¨ahler structures and bihermitian structures,we have the family of bihermitian structures ( J + t , J − t ) which is given by the action ofˇΓ ± t := Γ ± t (ˇ a ( t ) , ˇ b ( t )) of GL( T ). Since J ± t is integrable, we have π U − n +3 (cid:0) (ˇΓ ± t ) − d ˇΓ ± t (cid:1) = 0 . (3.10)Let Ω be a d -closed form of type ( n, 0) which is a local basis of K J = K J . Then as in theargument of proof of the proposition 1.4, we have d Γ ± t Ω ≡ k Γ ± t E ( t )Ω . Since d Ω = 0, the degree of E ( t ) is greater than or equal to 1. The condition [(Γ ± t ) [ i ] , J ] =0 (0 ≤ i < k ) implies that (Γ ± t ) [ i ] E ( t )Ω ∈ U − n +1 J . Thus we have d (Γ ± t ) [ k ] Ω = X i + j = k
The convergence As in the proposition 2.1, if there is a set of sections a , · · · a k − of CL which satisfies π U − n +3 (cid:0) e − a ( t ) de a ( t ) (cid:1) [ i ] = 0 , for all i < k, and k a ( t ) k s << k − K M ( t ), then there is a set of real sections b , · · · , b k ∈ ( L −J · L + J ⊕ L −J · L + J ) which satisfy the following equations: π U − n +3 (cid:0) e − Z ( t ) de Z ( t ) (cid:1) [ k ] = 0 (5.1) (cid:0) de Z ( t ) · ψ (cid:1) [ i ] = 0 , for all i ≤ k (5.2) k ˆ a k k s < K λM k (5.3) k ˆ b k k s < K M k (5.4)where ˆ a k is the section in the proposition 1.4 and M ( t ) is the convergent series in (1.12)with a constant λ and K is a positive constant and a positive constant K is determinedby λ, K . We also have an estimate of e Z ( t ) = e a ( t ) e b ( t ) in [10], k Z ( t ) k << k M ( t ) . Then γ k in (4.23) satisfies k γ k k s < k Γ + k ( a Thus it follows that ddt J − t | t =0 = [(Γ − t ) [1] , J ] = − β · ω + β · ω ) and the Kodaira-Spencerclass of deformations { J − t } is given by the class − β · ω ] ∈ H ( M. Θ). If the Kodaira-Spencer class does not vanish, then the deformations { J − t } is not trivial. Thus ( X, J − t ) isnot biholomorphic to ( X, J ) for small t = 0. A del Pezzo surface is by definition a smooth algebraic surface with ample anti-canonicalline bundle. A classification of del Pezzo surfaces are well known, they are C P × C P or C P or a surface S n which is the blow-up of C P at n points P , · · · , P n , (0 < n ≤ { P , · · · , P n } must be in general position to yield a del Pezzosurface. The following theorem is due to Demazure, [4] (see page 27), which shows themeaning of general position , Theorem . The following conditions are equivalent:(1) The anti-canonical line bundle of S n is ample(2) No three of Σ lie on a line, no six of Σ lie on a conic and no eight of Σ lie on a cubicwith a double point P i ∈ Σ (3) There is no curve C on S n with − K S n · C ≤ .(4) There is no curve C with C · C = − and K S n · C = 0 . emark . If three points lie on a line l , then the strict transform ˆ l of l in S isa ( − K S · ˆ l = 0. If six points belong to a conic curve C , then the stricttransform form ˆ C of C is again a ( − K S · C = 0. If eight points P · · · , P lie on a cubic curve with a double point P , then the strict transform ˆ C of C satisfiesˆ C ∼ π − C − E − E − · · · − E , where E i is the exceptional curve π − ( P i ). Then wealso have ˆ C = − K S · ˆ C = 0.Let D be a smooth anti-canonical divisor of S n which is given by the zero locus of asection β ∈ H ( S n , K − S n ). Since the anti-canonical bundle K − S n is regarded as the bundleof 2-vectors ∧ Θ and [ β, β ] S = 0 ∈ ∧ Θ on S n , every section β is a holomorphic Poissonstructure. On S n , we have the followings,dim H ( S n , Θ) = n − n = 5 , , , n < H ( S n , K − ) = 10 − n and H , ( S n ) = 1 + n. Further we have H ( S n , Θ) = { } , H ( S n , ∧ Θ) ∼ = H ( S n , − K S n ) = { } . Hencethe obstruction vanishes and we have deformations of generalized complex structuresparametrized by H ( S n , K − S n ) ⊕ H ( S n , Θ).In particular, if n ≥ 5, we have deformations of ordinary complex structures on S n Proposition . Let D be a smooth anti-canonical divisor given by the zero locus of β as above. Then there is a K¨aher form ω with the class [ β · ω ] = 0 ∈ H ( S n , Θ) . We also have H ( C P × C P , Θ) = 0 and H ( C P × C P , − K ) = 0.Thus we can apply our construction to every del Pezzo surface. From the main theorem0.1 together with the proposition 6.3, we have Proposition . Every del Pezzo surface admits deformations of bihermitian struc-tures ( J, J − t , h t ) with J − = J which satisfies ddt J − t | t =0 = − β · ω + β · ω ) , (6.1) for every K¨ahler form ω and every holomorphic Poisson structure β . Further, a del Pezzosurface S n ( n ≥ admits distinct bihermitian structures ( J, J − t , h t ) , that is, the complexmanifold ( X, J − t ) is not biholomorphic to ( X, J ) for small t = 0 . t = 0, J − t = ± J . We will give a proof of the proposition 6.3 in therest of this subsection.Let N D is the normal bundle to D in S n and i ∗ T S n the pull back of the tangent bundle T S n of S n by the inclusion i : D → S n . Then we have the short exact sequence,0 → T D → i ∗ T S n → N D → → H ( D, T D ) → H ( D, i ∗ T S n ) → H ( D, N D ) δ → H ( D, T D ) → · · · Since the line bundle N D is positive, H ( D, N D ) = { } and dim H ( D, N D ) is equal tothe intersection number D · D = 9 − n by the Riemann-Roch theorem. Since D is anelliptic curve, dim H ( D, T D ) = dim H ( D, T D ) = 1. Hence if follows that9 − n ≤ dim H ( D, i ∗ T S n ) ≤ − n. (6.2)Let I D be the ideal sheaf of D and O D the structure sheaf of D . Then we have the shortexact sequence 0 → I D → O S n → i ∗ O D → → I D ⊗ T S n → T S n → i ∗ O D ⊗ T S n → H p ( S n , i ∗ O D ⊗ T S n ) ∼ = H p ( S n , i ∗ ( O D ⊗ i ∗ T S n )) ∼ = H p ( D, i ∗ T S n ) , for p = 0 , , 2. From (6.3), we have the long exact sequence, H ( S n , T S n ) → H ( D, i ∗ T S n ) → H ( S n , I D ⊗ T S n ) j → H ( S n , T S n ) → · · · (6.4)Hence we obtain Lemma . The map j : H ( S n , I D ⊗ T S n ) → H ( S n , T S n ) is not the zero map . Proof. We have the exact sequence, · · · → H ( D, i ∗ T S n ) → H ( S n , I D ⊗ T S n ) j → H ( S n , T S n ) (6.5)From the Serre duality with I D = K S n , we have H ( S n , I D ⊗ T S n ) ∼ = H ( S n , Ω S n ) = { } and H ( S n , I D ⊗ T S n ) = H ( S n , Ω ) = 0. From the Riemann-Roch theorem, dim H ( S n , I D ⊗ T S n ) = n + 1. Then it follows from (6.2) thatdim H ( D, i ∗ T S n ) < dim H ( S n , I D ⊗ T S n )Note 10 − n < n + 1 for all n ≥ 5. Hence the map j is non zero.25 emark . Since n ≥ 5, we have H ( S n , T S n ) = { } . Applying the Serre dualitywith K S n = I D , we have H ( S n , T S n ) ∼ = H ( S n , I D ⊗ Ω ) = 0. From the Riemann-Roch,we obtain dim H ( S n , T S n ) = 2 n − β be a non-zero holomorphic Poisson structure S n with the smooth divisor D as the zero locus. Then β is regarded as a section of I D ⊗ ∧ Θ. Thus the section β ∈ H ( S n , I D ⊗ ∧ Θ) gives an identification,Ω ∼ = I D ⊗ T S n . Then the identification induces the isomorphismˆ β : H ( S n , Ω ) ∼ = H ( S n , I D ⊗ T S n ) . Let j be the map in the lemma 6.5. Then we have the composite map j ◦ ˆ β : H ( S n , Ω ) → H ( S n , Θ) which is given by the class [ β · ω ] ∈ H ( S n , T S n ) for [ ω ] ∈ H ( S n , Ω ). Proposition . The composite map j ◦ ˆ β : H ( S n , Ω ) → H ( S n , T S n ) is not thezero map. Proof. Since the map ˆ β is an isomorphism, ˆ β ( ω ) is not zero. It follows from lemma6.5 that the map j is non-zero. Hence the composite map j ◦ ˆ β is non-zero also. Proof. of lemma 6.3 The set of K¨ahler class is an open cone in H , ( S n , R ) ∼ = H ( S n R ). We have the non-zero map j ◦ ˆ β : H ( S n , C ) ∼ = H ( S n , Ω ) → H ( S n , Θ)for each β ∈ H ( S n , K − ) with { β = 0 } = D . It follows that the kernel j ◦ ˆ β is aclosed subspace and the intersection ker( j ◦ ˆ β ) ∩ H ( S n , R ) is closed in H ( S n , R ) whosedimension is strictly less than dim H ( S n , R ). Thus the complement in the K¨ahler cone (cid:8) [ ω ] : K¨ahler class | j ◦ ˆ β ([ ω ]) = 0 (cid:9) is not empty. Thus there is a K¨ahler form ω such that the class [ β · ω ] ∈ H ( S n , Θ) doesnot vanish for n ≥ Let M be a compact complex surface with canonical line bundle K M . We shall givesome vanishing theorems of the cohomology groups H ( M, − K M ) and H ( M, Θ) on acompact smooth complex surface M , which are the obstruction spaces to deformations ofgeneralized complex structures starting from the ordinary one ( X, J J ). The following ispractical to show the vanishing of H ( M, − K M ).26 roposition . Let M be a compact complex surface with H ( M, O M ) = 0 . If − K M = m [ D ] for a irreducible, smooth curve D with positive self-intersection number D · D > and a positive integer m , then H ( M, K nM ) = 0 for all integer n . The proposition is often used in the complex geometry. For completeness, we give aproof. Proof. Let I D be the ideal sheaf of the curve D . Then we have the short exactsequence, 0 → I D → O M → j ∗ O D → 0, where j : D → X . Then we have the exactsequence, H ( M, O M ) → H ( M, j ∗ O D ) δ → H ( M, I D ) → H ( M, O M )It follows that the coboundary map δ is a 0-map. Thus from H ( M, O M ) = 0, wehave H ( M, I D ) = H ( M, − [ D ]) = 0. We use the induction on k . We assume that H ( M, I kD ) = H ( M, − k [ D ]) = 0 for a positive integer k . The short exact sequence0 → I k +1 D → I kD → j ∗ O D ⊗ I kD → H ( M, j ∗ O D ⊗ I kD ) → H ( M, I k +1 D ) → H ( M, I kD ) . By the projection formula, we have H ( M, j ∗ O D ⊗ I kD ) = H ( D, − k [ D ] | D ). Since D · D > 0, it follows that the line bundle − k [ D ] | D is negative and then H ( D, − k [ D ] | D ) = H ( M, I kD ) = 0. It implies that H ( M, I k +1 D ) = H ( M, − ( k + 1)[ D ]) = 0. Thus by theinduction, we have H ( M, − nD ) = 0 for all positive integer n . Applying the Serre duality,we have H ( M, − nD ) ∼ = H ( M, ( n − m ) D ) = 0. Thus H ( M, nD ) = 0 for all integer n .Then the result follows since H ( M, K n ) = H ( M, − ( nm ) D ) = 0.The author also refer to the standard vanishing theorem. If D = P i a i D i is a Q -divisoron M , where D i is a prime divisor and a i ∈ Q . Let ⌈ a i ⌉ be the round-up of a i and ⌊ a i ⌋ the round-down of a i . Then the fractional part { a i } is a i − ⌊ a i ⌋ . Then the round-up andthe round-down of D is defined by ⌈ D ⌉ = X i ⌈ a i ⌉ D i , ⌊ D ⌋ = X i ⌊ a i ⌋ D i and { D } = P i { a i } D i is the fractional part of D . A divisor D is nef if one has D · C ≥ C . A divisor D is nef and big if in addition, one has D > 0. We shall usethe following vanishing theorem. The two dimensional case is due to Miyaoka and thehigher dimensional cases are due to Kawamata and Viehweg Theorem . Let M be a smooth projective surface and D a Q -divisor on M suchthat(1) supp { D } is a divisor with normal crossings,(2) D is nef and big.Then H i ( M, K M + ⌈ D ⌉ ) = 0 for all i > . − K M = mD is nef and big divisor where D is smooth for m > 0. Then applyingthe theorem, we have H i ( M, − K M ) ∼ = H i ( M, K M − K M ) = 0 , for all i > H ( M, Θ). Applying the Serreduality theorem, we have H ( M, Θ) ∼ = H ( M, Ω ⊗ K M )If − K M is an effective divisor [ D ], then K M is given by the ideal sheaf I D of D . The shortexact sequence: 0 → Ω ⊗ I D → Ω → Ω ⊗ O D → → H ( M, Ω ⊗ K M ) → H ( M, Ω ) . Hence we have Proposition . if M is a smooth surface with effective anti-canonical divisorsatisfying H ( M, Ω ) = 0 , then we have the vanishing H ( M, Θ) = 0 . Proposition . Let M be a K¨ahler surface with a K¨ahler form ω and a non-zeroPoisson structure β ∈ H ( M, ∧ Θ) . Let D be the divisor defined by the section β . Ifthere is a curve C of M with C ∩ supp D = ∅ , then the class [ β · ω ] ∈ H ( M, Θ) does notvanish. Proof. Since β is not zero on the complement M \ D , there is a holomorphic sym-plectic form ˆ β on the complement. The symplectic form ˆ β gives the isomorphism Θ ∼ = Ω on M \ D which induces the isomorphism between cohomology groups H ( M \ D, Θ) ∼ = H ( M \ D, Ω ). Then the restricted class [ β · ω ] | M \ D corresponds to the K¨ahler class[ ω ] | M \ D ∈ H ( M \ D, Ω ) ∼ = H , ( M \ D ) under the isomorphism. Since there is the curve C on the complement M \ D and ω is a K¨ahler form, the class [ ω | C ] ∈ H , ( C ) does notvanish. Then it follows that the class [ ω ] | M \ D ∈ H ( M \ D, Ω ) does not vanish. It impliesthat [ β · ω ] | M \ D does not vanish also. Thus we have that the class [ β · ω ] ∈ H ( M, Θ) doesnot vanish. F and F Let F be the projective space bundle of T ∗ C P ⊕ O C P , F = P ( T ∗ C P ⊕ O C P ) . 28e denote by E + and E − the sections of F with positive and negative self-intersectionnumbers respectively. An anti-canonical divisor of F is given by 2 E + , while the section E − with E − · E − = − E + ∩ E − = ∅ . Thus we have thenon-vanishing class [ β · ω ] ∈ H ( F , Θ), where β is a section of − K with the divisor 2 E + .(Note that the canonical holomorphic symplectic form ˆ β on the cotangent bundle T ∗ C P which induces the holomorphic Poisson structure β . The structure β can be extended to F which gives the anti-canonical divisor 2[ E + ].) Proposition . The class [ β · ω ] ∈ H ( F , Θ) does not vanish for every K¨ahlerform ω on F . Proof. The result follows from the proposition 6.11.On the surface F , the anti-canonical line bundle of F is 2 E + and H ( F , O F ) = 0.Hence from the proposition 6.8, we have the vanishing H i ( F , − K X ) = { } for all i > 0. Since the surface F is simply connected, it follows from the proposition 6.10 that H ( F , Θ) = 0. Hence the obstruction vanishes and we can apply our main theorem . Itis known that every non-trivial small deformation of F is C P × C P . Thus we have Proposition . Let ( X, J ) be the Hirtzebruch surface F as above. Then there isa family of deformations of bihermitian structures ( J + t , J − t , h t ) with J + t = J − = J suchthat ( X, J − t ) is C P × C P for small t = 0 . Let F e be the projective space bundle P ( O ⊕ O ( − e )) over C P with e > 0. There isa section b with b = − e , which is unique if e > 0. Let f be a fibre of F e . Then − K isgiven by 2 b + ( e + 2) f , which is an effective divisor. Thus from the proposition 6.10, wehave H ( F e , Θ) = { } . P − ( F e ) = dim H ( F e , K − ) is listed in the table 7.1.1 of [24], P − ( F e ) = e = 0 , e = 2 e + 6 e ≥ K is given by the ideal sheaf I D for the effective divisor D = 2 b + ( e + 2) f , Itfollows from the Serre duality that H ( F e , K − ) = H ( F e , I D ) = { } . Thus applying theRiemann-Roch theorem, we obtaindim H ( F e , K − ) = e − , for e ≥ 3. In the case e = 3, we have H ( F , K − ) = H ( F , Θ) = { } . Thus from thetheorem 0.1, we have 29 roposition . The Hirtzebruch surface F admits deformations of bihermitianstructures ( J, J − t , h t ) with J − t = ± J for small t = 0 . We can generalized our discussion of F to the projective space bundle of T ∗ M ⊕ O M over a compact K¨ahler manifold M . Then we also have the Poisson structure β and asin the proposition 6.11, it is shown that the class [ β · ω ] does not vanish. Thus we havethe deformations of bihermitian structures from the stability theorem [11]. For the onesas in the theorem 0.1, we need to show the vanishing of the obstruction. Note that theobstruction space does not vanish in general.If M is a Riemannian surface Σ g of genus g ≥ 1, then the projective space bundle iscalled a ruled surface of degree g . it is known that small deformations of any ruled surfaceof degree g ≥ Proposition . Let ( X, J ) be a ruled surface P ( T ∗ Σ g ⊕ O Σ g ) with degree g ≥ .Then there is a family of non-trivial bihermitian structures ( J + t , J − t , h t ) such that J = ± J ± t and ( X, J ± t ) is a ruled surface for small t . We shall consider the blow-up of C P at r points which are not in general position. Wefollow the construction as in [4], (see page 36). We have a finite set Σ = { x , · · · , x r } and X (Σ) obtained by successive blowing up at Σ, X (Σ) → X (Σ r − ) → · · · → X (Σ ) → C P , At first X (Σ ) is the blow-up of C P at a point x ∈ C P and we have Σ i = { x , · · · , x i } and X (Σ i +1 ) is the blow-up of X (Σ i ) at x i +1 ∈ X (Σ i ). Let E i be the divisor given bythe inverse image of x i ∈ X (Σ i − ). If Γ is an effective divisor on C P , one notes thatmult( x i , Γ) the multiplicity of x i on the proper transform of Γ in X (Σ i − ), and one saysthat Γ passes through x i if mult( x i , Γ) > 0. Define ˆ E , · · · , ˆ E r by recurrence as follows,On X (Σ ), one put ˆ E = E ; on X (Σ ), ˆ E is a proper transform of the previous E and one also put ˆ E = E ; on X (Σ ), ˆ E and ˆ E are the proper transform of previousˆ E and ˆ E respectively and ˆ E = E . Then ˆ E , · · · , ˆ E r are irreducible components of E + · · · + E r .We assume that the following condition on Σ,(*) For each i = 1 , · · · , r , a point x i ∈ X (Σ i − ) does not belong to a irreducible curve ˆ E j with self-intersection number − ≤ j ≤ i − x i ∈ X (Σ i − ) belongs to a irreducible curve ˆ E j with self-intersection num-ber − 2, then the proper transform of ˆ E j becomes a curve with self-intersection number − 3. If there is a rational curve with self-intersection number − X (Σ) is not nef. Definition . A set of points Σ is in almost general position if Σ satisfies thefollowing:(1) Σ satisfies the condition (*)(2) No line passes through 4 points of Σ(3) No conic passes through 7 points of ΣWe call X (Σ) a degenerate del Pezzo surface if Σ is in almost general position. Notethat if Σ is in general position, Σ is in almost general position. In [4], the followingtheorem was shown, Theorem . [4] The following conditions are equivalent:(1) Σ is in almost general position(2) The anti-canonical class of X (Σ) contains a smooth and irreducible curve D .(3) There is a smooth curve of C P passing all points of Σ .(4) H ( X (Σ) , K nX (Σ) ) = { } for all integer n (5) − K X Σ · C ≥ for all effective curve C on X (Σ) and in adition, if − K X (Σ) · C = 0 ,then C · C = − . Then from (2) there is a smooth anti-canonical divisor on a degenerate del Pezzosurface and we have H ( X (Σ) , O X ) = 0. Hence from the proposition 6.8, we have thevanishing H i ( X (Σ) , − K X ) = 0, for all i > 0. A degenerate del Pezzo surface X (Σ) satisfies H ( X (Σ) , Ω ) = 0. Then it follows from the proposition 6.10 that H ( X (Σ) , Θ) = 0.Let X (Σ) be a degenerate del Pezzo surface which is not a del Pezzo surface, that is,the anti-canonical class of X (Σ) is not ample. Then from (5), there is a ( − C with K X (Σ) . · C = 0. Then it follows that C is a C P . Thus we contract ( − β be a section of − K X with the smoothdivisor D as the zero set. We denote by J the complex structure of the del Pezzo surface X (Σ). From the theorem 0.1, we have Theorem . A degenerate del Pezzo surface admits deformations of distinct bi-hermitian structures ( J, J − t , h t ) with J − = J and J − t = ± J for small t = 0 , that is, ddt J − t | t =0 = − β · ω + β · ω ) , and the complex structure J − t is not equivalent to J of X (Σ) under diffeomorphisms for small t = 0 , where ω is a K¨ahler form. roof. If X (Σ) is a del Pezzo surface, we already have the result. If X (Σ) is not a delPezzo but a degenerate del Pezzo, we still have H ( X (Σ) , Θ) = H ( X (Σ) , K − ) = { } .Thus we have deformations of bihermitian structures as in the theorem 0.1. It is sufficientto show that the class [ β · ω ] does not vanish. Since K · C = 0, the line bundle K | C → C ∼ = C P is trivial. If there is a point P ∈ D ∩ C , then β ( P ) = 0 and it follows that β | C ≡ D is smooth, we have D = C . However D · D = 9 − r and D · C = − K · C = 0. Thus D ∩ C = ∅ . Then applying the proposition 6.11, we obtain [ β · ω ] = 0 ∈ H ( X (Σ) , Θ). We shall discuss an analog of the Kuranishi family of deformations of generalized com-plex structures. The deformation theory of generalized complex structures was alreadyobtained in [12] by using the implicit function theorem. For the completeness of this paper,we will give the different construction of deformations of generalized complex structuresby using the power series. Our method explicitly shows that the deformations familydepends holomorphically on the parameter t and we can also have an estimate of theconvergent series as in section 1.Let ( X, J ) be a compact generalized complex manifold and L := L J the Lie algebroidbundle as before which gives the decomposition, ( T ⊕ T ∗ ) C = L ⊕ L . Note that theobstruction space H ( ∧ • L J ) does not necessary vanish. Even in the case we obtain thefamily of deformations which is parametrized by an analytic set. We fix a metric on X and consider the adjoint d ∗ L , where d L is the derivative of the complex, · · · d L → ∧ k L J d L → ∧ k +1 L J d L → · · · . We also denote by G L the Green operator of the Laplacian △ L := d L d ∗ L + d ∗ L d L .Let { η i } mi =1 be a basis of the Harmonic forms H ( L ) ∼ = H ( L ). As in (1.18) we alsohave the convergent series ε ( t ) which is a unique solution of ε ( t ) = ε ( t ) − d ∗ L G L [ ε ( t ) , ε ( t )] S , (7.1)where ε ( t ) = P mi =1 η i t i and t = ( t , , · · · , t m ) ∈ C m . Note that ε ( t ) is not a section withone variable but one with several variables t = ( t , , · · · , t m ). The convergent series ε ( t )is determined by the first term ε ( t ). The harmonic component of [ ε ( t ) , ε ( t )] S is denotedby H ([ ε ( t ) , ε ( t )] S ) ∈ H ( ∧ • L ). We define an analytic set A by A = { t ∈ C m (cid:12)(cid:12) | t | < α, H ([ ε ( t ) , ε ( t )] S ) = 0 } where α is a sufficiently small constant. 32 roposition . We have a family of generalized complex structures {J t } which isparametrised by the analytic set A . Our proof is almost same as in the one of complex deformations and we use the similarnotation as in [16]. Proof. It suffices to show that for a fixed ε ( t ), the ε ( t ) in (7.1) satisfies the Maurer-Cartan equation if and only if H ([ ε ( t ) , ε ( t )] S ) = 0. If ε ( t ) is a solution of the Maurer-Cartan equation, d L ε ( t ) + 12 [ ε ( t ) , ε ( t )] S = 0 . Then it follows that the harmonic part H ([ ε ( t ) , ε ( t )] S ) vanishes. Conversely, we assumethat H ([ ε ( t ) , ε ( t )] S ) = 0. Let Ψ = d L ε ( t ) + [ ε ( t ) , ε ( t )] S ∈ ∧ L . It follows from (7.1)that that d L ε ( t ) = − d L d ∗ L G L [ ε ( t ) , ε ( t )] S . Then applying the Hodge decomposition to[ ε ( t ) , ε ( t )] S , we have 2Ψ = − d L d ∗ L G L [ ε ( t ) , ε ( t )] S + [ ε ( t ) , ε ( t )] S (7.2)= H ([ ε ( t ) , ε ( t )] S ) + d ∗ L d L G L [ ε ( t ) , ε ( t )] S (7.3)= d ∗ L d L G L [ ε ( t ) , ε ( t )] S (7.4)By using the proposition 1.2 and substituting d L ε ( t ) = Ψ − [ ε ( t ) , ε ( t )] S , we haveΨ = d ∗ L G L [ d L ε ( t ) , ε ( t )] S (7.5)= d L G L [Ψ , ε ( t ) ] S − d L G L (cid:2) [ ε ( t ) , ε ( t )] S , ε ( t ) (cid:3) S (7.6)= d L G L [Ψ , ε ( t ) ] S . (7.7)We use the Sobolev norm k k s and the elliptic estimate, k Ψ k s 1, it followsthat Ψ = 0. Hence ε ( t ) satisfies the Maurer-Cartan equation. We will give a short explanation of the Schouten bracket and the proposition 1.2. Ourdefinition of the Schouten bracket is called the Dervied bracket construction [19]. Let( X, J ) be a generalized complex manifold with th decomposition ( T ⊕ T ∗ ) C = L J ⊕ L J .We denote by ∧ • L J the skew-symmetric forms of L J , which acts on differential forms33 • T ∗ by the spin representation. Let K J be the canonical line bundle which is given by K J = { φ ∈ ∧ • T ∗ | L J · φ = 0 } . Then the space of differential forms is decomposed intoirreducible representations: ∧ • T ∗ = ⊕ ni =0 U − n + p , where each component U − n + p is givenby ∧ p L J · K J . For a section ε ∈ ∧ p L J , we denote by | ε | := p the degree of ε . Theexterior derivative d is decomposed into d = ∂ + ∂ , where ∂ : U − n + p → U − n + p − and thecomplex conjugate ∂ : U − n + p → U − n + p +1 . We consider ε ∈ ∧ • L J is an operator from K J to U − n + | ε | by the spin representation of ∧ • L J on ∧ • T ∗ . For ε , ε ∈ ∧ • L J , we define agraded bracket [ , ] G by [ ε , ε ] G = ε ε − ( − | ε | | ε | ε ε . Let A be a differential operatoracting on ∧ • T ∗ . If A : U − n + i → U − n + i + a , for all i , A is an operator of degree a = | A | .For operators A, B of degree | A | and | B | , we also have the graded bracket:[ A, B ] G := AB − ( − | A | | B | BA. The exterior derivative d admits the decomposition d = ∂ + ∂ , where ∂ and ∂ areoperators of degree 1 and − d is an operator of odd degree and thegraded commutator with ε ∈ ∧ L J is given by Dε := [ d, ε ] G = dε − ( − | ε | εd. Then we define Schouten bracket [ ε , , ε , ] S ∈ ∧ | ε | + | ε |− L J by[ ε , ε ] S := [ Dε , ε ] G = [ [ d, ε ] G , ε ] G = [ [ ∂, ε ] G , ε ] G , (8.1)where [ [ ∂, ε ] G , ε ] G = 0 and [ ε , , ε , ] S ∈ ∧ | ε | + | ε |− L J . Let d L be the derivative of theLie algebroid L J . Then we have d L ε = [ ∂, ε ] G ∈ ∧ p +1 L J (8.2)(Refer to [9].) In fact, since we have [ ∂, ε ] G f φ = f [ ∂, ε ] G + [ ∂f, ε ] G φ = f [ ∂, ε ] G , the oper-ator [ ∂, ε ] G is regarded as an element of Hom( K J , U − n + | ε | +1 ) and since [[ ∂, ε ] G , ε ] G φ = 0for φ ∈ K J and ε ∈ ∧ • L J , the commutator [ ∂, ε ] G is also an element of ∧ | ε | +1 L J underthe isomorphism ∧ p L J ∼ =Hom( K J , U − n + p ), which is given by the spin representation.Then we obtain an isomorphism between two complexes:( ∧ • L J , d L ) ∼ = ( U − n + • ⊗ K − J , [ ∂, ] G )In fact we have (cid:2) ∂, [ ∂, ε ] G (cid:3) G = (cid:2) ∂, ( ∂ε − ( − | ε | ε∂ ) (cid:3) G (8.3)= ∂∂ε − ( − | ε | ∂ε∂ − ( − | ε | +1 ∂ε∂ − ε∂∂ (8.4)=0 (8.5)From now we identify d L ε with [ ∂, ε ] G .We have the following relations of the graded bracket.34 emma . [ A, B ] G = − ( − | A | | B | [ B, A ] G , (8.6) the Jacobi identity of the graded bracket holds (cid:2) [ A, B ] G , C (cid:3) G ( − | A || C | + (cid:2) [ B, C ] G , A (cid:3) G ( − | B || A | + (cid:2) [ C, A ] G , B (cid:3) G ( − | C || B | = 0 (8.7) Proof. These follows from a direct calculations.We also have the following three relations of the Schouten bracket, Lemma . [ ε , ε ] S = ( − | ε || ε | [ ε , ε ] S Lemma . d L [ ε , ε ] S = [ d L ε , ε ] S + ( − | ε | [ ε , d L ε ] S Lemma . (cid:2) [ ε , ε ] S , ε (cid:3) S ( − | ε || ε | + (cid:2) [ ε , ε ] S , ε (cid:3) S ( − | ε || ε | + (cid:2) [ ε , ε ] S , ε (cid:3) S ( − | ε || ε | = 0We shall show that every lemma follows from (8.2) and lemma 8.1. Proof of lemma 8.2 for ε , ε ∈ ∧ • L J , we have D [ ε , ε ] G = [ Dε , ε ] G + ( − | ε | [ ε , Dε ] G = 0 (8.8)Since [ ε , ε ] G = 0, we have [ Dε , ε ] G + ( − | ε | [ ε , Dε ] G = 0 . Since [ ε , Dε ] G = − ( − | ε | ( | ε | +1) [ Dε , ε ] G , we obtain[ Dε , ε ] G = ( − | ε | | ε | [ Dε , ε ] G It implies that [ ε , ε ] S = ( − | ε | | ε | [ ε s , ε ] S . Proof of lemma 8.3 From (8.1) we have d L [ ε , ε ] S = (cid:2) ∂, [ ε , ε ] S (cid:3) G = (cid:2) ∂, (cid:2) Dε , ε ] G (cid:3) G (8.9)Applying the lemma 8.1, we have( − | ε | d L [ ε , ε ] S = (cid:2) [ Dε , ε ] G , ∂ (cid:3) G ( − − | ε | + | ε |− ( − | ε | (8.10)= (cid:2) [ Dε , ε ] G , ∂ (cid:3) G ( − − | ε |− (8.11)= (cid:2) [ ε , ∂ ] G , Dε (cid:3) G ( − | ε | ( | ε |− (8.12)+ (cid:2) [ ∂, Dε ] G , ε (cid:3) G ( − | ε | (8.13)35rom the lemma 8.1 , we also have[ ∂, Dε ] G =[ Dε , ∂ ] G ( − − | ε |− (8.14)= (cid:2) [ ∂, ε ] G , ∂ (cid:3) G ( − − | ε |− (8.15)= (cid:2) [ ε , ∂ ] G , ∂ (cid:3) G ( − | ε |− ( − | ε | (8.16)+ (cid:2) [ ∂, ∂ ] G , ε ] G ( − | ε |− ( − | ε | (8.17)Since [ ∂, ∂ ] G = ∂∂ + ∂∂ = 0, we have[ ∂, Dε ] G = (cid:2) [ ε , ∂ ] G , ∂ ] G ( − 1) (8.18)= (cid:2) [ ∂, ε ] G , ∂ (cid:3) G ( − | ε | (8.19)= (cid:2) ∂, [ ∂, ε ] G (cid:3) G ( − | ε | ( − − | ε | +1 (8.20)= Dd L ε (8.21)Substituting them, we obtain( − | ε | d L [ ε , ε ] S = (cid:2) [ ∂, ε ] G , Dε (cid:3) G ( − | ε | ( | ε |− ( − − | ε | (8.22)+ (cid:2) Dd L ε , ε (cid:3) G ( − | ε | (8.23)(8.24)Hence we have d L [ ε , ε ] S = (cid:2) Dε , [ ∂, ε ] G (cid:3) G ( − | ε | ( | ε |− ( − − ( | ε |− 1) ( | ε | +1) (8.25)+ (cid:2) Dd L ε , ε (cid:3) G (8.26)= (cid:2) Dd L ε , ε (cid:3) G + ( − | ε | (cid:2) Dε , [ ∂, ε ] G (cid:3) G (8.27)=[ d L ε , ε ] S + ( − | ε | [ ε , d L ε ] S (8.28) Proof of lemma 8.4 For ε , ε , ε ∈ ∧ • L J , it follow from (8.8) that one have (cid:2) [ ε , ε ] S , ε (cid:3) S = (cid:2) D [ Dε , ε ] G , ε (cid:3) G =[ [ Dε , Dε ] G , ε ] G ( − ( | ε | +1) (cid:2) [ ε , ε ] S , ε (cid:3) S = (cid:2) D [ Dε , ε ] G , ε (cid:3) G = (cid:2) [ Dε , ε ] G , Dε (cid:3) G ( − ( | ε | + | ε | ) (cid:2) [ ε , ε ] S , ε (cid:3) S = (cid:2) D [ Dε , ε ] G , ε (cid:3) G = (cid:2) D [ ε , Dε ] G , ε (cid:3) G ( − ( | ε | +1) = (cid:2) [ ε , Dε ] G , Dε (cid:3) G ( − ( | ε | +1) ( − ( | ε | + | ε | ) (cid:2) [ ε , ε ] S , ε (cid:3) S ( − | ε || ε | =[ [ Dε , Dε ] G , ε ] G ( − ( | ε | +1) ( − | ε || ε | (8.29) (cid:2) [ ε , ε ] S , ε (cid:3) S ( − | ε || ε | = (cid:2) [ Dε , ε ] G , Dε (cid:3) G ( − ( | ε | + | ε | ) ( − | ε || ε | (8.30) (cid:2) [ ε , ε ] S , ε (cid:3) S ( − | ε || ε | = (cid:2) [ ε , Dε ] G , Dε (cid:3) G ( − ( | ε | +1) ( − ( | ε | + | ε | ) ( − | ε || ε | (8.31)Multiplying ( − ( | ε |−| ε |− , we have( − ( | ε |−| ε |− (cid:2) [ ε , ε ] S , ε (cid:3) S ( − | ε || ε | =[ [ Dε , Dε ] G , ε ] G ( − ( | ε | +1) | ε | (8.32)( − ( | ε |−| ε |− (cid:2) [ ε , ε ] S , ε (cid:3) S ( − | ε || ε | = (cid:2) [ Dε , ε ] G , Dε (cid:3) G ( − ( | ε | +1)( | ε | +1) (8.33)( − ( | ε |−| ε |− (cid:2) [ ε , ε ] S , ε (cid:3) S ( − | ε || ε | = (cid:2) [ ε , Dε ] G , Dε (cid:3) G ( − | ε | ( | ε | +1) (8.34)We apply the Jacobi identity of the graded bracket [ , ] G (cid:2) [ A, B ] G , C (cid:3) G ( − | A || C | + (cid:2) [ B, C ] G , A (cid:3) G ( − | B || C | + (cid:2) [ C, A ] G , B (cid:3) G ( − | C || B | = 0(8.35)Then we have the Jacobi identity of the Schouten bracket (cid:2) [ ε , ε ] S , ε (cid:3) S ( − | ε || ε | + (cid:2) [ ε , ε ] S , ε (cid:3) S ( − | ε || ε | + (cid:2) [ ε , ε ] S , ε (cid:3) S ( − | ε || ε | = 0 . 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