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Unobstructed deformations of generalized complex structures 5 induced by C∞ logarithmic symplectic structures 1 0 and logarithmic Poisson structures 2 n a Ryushi Goto J 4 Department of Mathematics, Graduate School of Science, OsakaUniversity 1 [email protected] ] G D . h t Abstract a m WeshallintroducethenotionofC∞logarithmicsymplecticstructuresonadiﬀerentiablemanifold [ which is an analog of the one of logarithmic symplectic structures in the holomorphic category. We 1 show that the generalized complex structure induced by a C∞ logarithmic symplectic structure has v unobstructed deformations which are parametrized by an open set of the second de Rham cohomol- 8 ogy group of the complement of type changing loci if the type changing loci are smooth. Complex 9 surfaceswithsmootheﬀectiveanti-canonicaldivisorsadmitunobstructeddeformationsofgeneralized 3 3 complex structures such as del pezzo surfaces and Hirzebruch surfaces. We also give some calcula- 0 tions of Poisson cohomology groups on these surfaces. Generalized complex structures on the m 1. connected sum (2k 1)CP2#(10k 1)CP2 as in [CG1], [GH] are induced by C∞ logaritJhmic sym- − − 0 plectic structures modulo the action of b-ﬁelds and it turns out that generalized complex structures 5 haveunobstructed deformations of dimension 12k+2m 3. 1 Jm − : v i Contents X r a 1 Introduction 2 2 Generalized complex structures 3 3 Deformation theory of generalized complex structures 4 4 Log symplectic structure ωC in holomorphic category and generalized complex structure 6 φ J 5 Lie algebroid cohomology groups of and logarithmic Poisson structure β 7 φ J 6 Unobstructed deformations of generalized complex structures induced from C ∞ logarithmic symplectic structures 8 7 Logarithmic deformations of 9 φ J 2010Mathematics Subject Classiﬁcation. Primary53C55;Secondary 32G05. Keywords and phrases. ∗PartlysupportedbytheGrant-in-AidforScientiﬁcResearch(B),JapanSocietyforthePromotionofScience. 1 8 Proof of main theorems 10 9 Generalized complex structures on 4-manifolds 11 9.1 Non-degenerate, purespinors of even typeon 4-manifolds . . . . . . . . . . . . . . . . 11 9.2 Unobstructed deformations of generalized complex structureson Poisson surfaces . . . 11 9.3 C∞logarithmictransformationsandunobstructeddeformationsofgeneralizedcomplex structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1 Introduction Generalized complex structures are mixed geometric structures building a bridge between complex geometryandrealsymplecticgeometry. Bothcomplexstructuresandrealsymplecticstructuresgive risetogeneralized complexstructuresofspecialclasses. Howeverageneralized complexstructureon amanifold can admit thetypechangingloci on which thetypeofthestructurecan change form the onefrom a real symplectic structure to theone from a complex structure. AcomplexsurfaceSwithanontrivialholomorphicPoissonstructureβhasageneralizedcomplex structure with type changing loci at zeros of β. It is striking that (3k 1)CP3#(10k 1)CP2 β J − − does not admit complex structures and real symplectic structures, but (3k 1)CP3#(10k 1)CP2 − − has generalized complex structures with m typechanging loci [CG1], [GH]. m J ThedeformationcomplexofageneralizedcomplexstructureisgivenbytheLiealgebroidcomplex ( •L ,dL). ThenthespaceofinﬁnitesimaldeformationsisthesecondcohomologygroupH2( •L ) ∧ J ∧ J andtheobstruction space isthethirdcohomology group H3( •L ). Then theKuranishifamilies of ∧ J generalized complex structures are constructed [Gu1]. A C∞ logarithmic symplectic structure ωC on a manifold M along a submanifold D of real codi- mension 2 is a complex 2-form which is given bythe following on a neighborhood of D, ωC = dzz11 ∧dz2+dz3∧dz4+···+dz2m−1∧dz2m, where (z1, ,z2m) are complex coordinates and D = z1 = 0 . On a neighborhood of the com- ··· { } plement M D, ωC is a smooth 2-form b+√ 1ω where b is a d-closed 2-form and ω denotes a real \ − symplecticstructure. (Notethatwedonotassume thatM isacomplex manifold.) Theexponential φ:=eωC gives rise to a generalized complex structure φ J One of the purposes of the paper is to show that the generalized complex structure induced φ J byaC∞ logarithmic symplectic structureωC hasunobstructeddeformations which are given bythe second cohomology group of the complement of type changing loci if type changing loci are smooth (see Theorem 6.2 and Theorem 6.3). Ourunobstructedness theorems may be regarded as an analog oftheunobstructednesstheoremsofCalabi-YauandhyperK¨ahlermanifolds. Howeverwedonotuse theHodge theory and the ∂∂-lemma but our method is rathertopological. Thegeneralizedcomplexstructure onaPoissonsurfaceS and onthe(3k 1)CP3#(10k β m J J − − 1)CP2 are induced by C∞ logarithmic symplectic structures modulo the action of b-ﬁelds. Thus we can apply our unobstructedness theorems to these generalized complex structures (see Theorem 9.2 and Theorem 9.19). The paper is organized as follows. In Section 2 we give a short explanation of generalized com- plex structures and in Section 3 we also provide fundamental notions of deformations of generalized complexstructures. Itisremarkablethat Poisson cohomology groupsofaPoisson structureβ which arethehypercohomology groupsof thePoisson complex coincidewith theLiealgebroid cohomology groups of the generalized complex structure induced by β. In Section 4, we discuss the gener- β J alized complex structures induced from a logarithmic symplectic structure along a smooth divisor D on a complex manifold X which is the dual of a holomorphic Poisson structure β. In Section 5, we show that the Poisson cohomology groups of β are isomorphic to the de Rham cohomology 2 groups of the complement X D. In Section 6, we introduce C∞ logarithmic symplectic structures \ andshow ourmain theorems. InSection 7, we discusslogarithmic deformations. Proofs of ourmain Theorems are given in Section 8. In Section 9, the unobstructedness theorem can be applied to a complex surface with smooth anti-canonical divisor to obtain unobstructed deformations. We cal- culate Poisson cohomology groups and the de Rham cohomology groups of the complements. It is intriguing that there are diﬀerences between these two cohomology groups when D has singularities (seeRemark9.4). InSection 9.3, weexplain aconstruction ofgeneralized complex structures on m J (3k 1)CP3#(10k 1)CP2 bylogarithmictransformationswithmtypechangingloci. Weapplyour − − theorems to generalized complex structures on (3k 1)CP3#(10k 1)CP2 to obtain unobstructed − − deformations of generalized complex structures (see Theorem 9.19). 2 Generalized complex structures LetTM bethetangentbundleonadiﬀerentiablemanifold ofdimension 2nandT∗M thecotangent bundle of M. The symmetric bilinear form , on the direct sum TM T∗M is deﬁned by v+ h i ⊕ h ξ,u+ηi = 21(ξ(u)+η(v)), where u,v ∈ TM,ξ,η ∈ T∗M. Then the symmetric bilinear form h, i yieldstheﬁbrebundleSO(TM T∗M) withﬁbrethespecial orthogonal group. Asection ofbundle ⊕ SO(TM T∗M) is an endomorphism of TM T∗M preserving , and its determinant is equal to ⊕ ⊕ h i one. If a section of SO(TM T∗M) satisﬁes 2 = id, then J is called an almost generalized J ⊕ J − complexstructurewhichgivesthedecomposition (TM T∗M)C =L L intoeigenspaces,where ⊕ J ⊕ J L istheeigenspaceofeigenvalue√ 1andL isthecomplexconjugateofL . TheCourantbracket J − J J is deﬁnedby 1 [u+ξ,v+η]co =[u,v]+Luη−Lvξ− 2(diuη−divξ), where[u,v]denotesthebracketofvectorﬁeldsuandv and η and ξ aretheLiederivativesand v u L L i η and i ξ stand for the interior products. u v If L is closed with respect to the Courant bracket, then is a generalized complex structure, J J that is, [e1,e2]co L for all e1,e2 L . ∈ J ∈ J The direct sum TM T∗M acts on diﬀerential forms bythe interior and exterior products, ⊕ e φ=(v+η) φ=i φ+η φ, v · · ∧ where e=v+η and v TM and η T∗M and φ is a diﬀerential forms. Then it turnsout that ∈ ∈ e e φ=(v+η) (v+η) φ=η(v)φ= e,e φ. · · · · h i SinceitistherelationoftheCliﬀordalgebrawithrespectto , ,weobtaintheactionoftheCliﬀord h i algebra bundleCl(TM T∗M) on diﬀerential forms which is the spin representation. ⊕ WedeﬁnekerΦ:= E (TM T∗M)C E Φ=0 for a diﬀerential form Φ even/oddT∗M. If { ∈ ⊕ | · } ∈∧ kerΦ is maximal isotropic, i.e., dimCkerΦ=2n, then Φ is called a pure spinor of even/odd type. ApurespinorΦisnondegenerateifkerΦ kerΦ= 0 ,i.e.,(TM T∗M)C =kerΦ kerΦ. Then ∩ { } ⊕ ⊕ a nondegenerate, pure spinor Φ •T∗M gives an almost generalized complex structure Φ which ∈ ∧ J satisﬁes +√ 1E, E kerΦ JΦE = √−1E, E ∈kerΦ − − ∈ Conversely,ageneralizedcomplexstructure arisesas Φforanondegenerate,purespinorΦwhichis J J uniqueuptomultiplicationbynon-zerofunctions. Thusthereisaonetoonecorrespondencebetween almost generalized complex structures and non-degenerate, pure spinors modulo multiplication by tnhoen-nzoenro-dfeugnecnteiroantse.,pTuhreecsapninoonricΦa.l linΦeisbuinntdelgeraKbJleΦifoafnJdΦonislythifedcΦom=pElexΦlinfoerbEundl(eTgMenerTat∗eMd)bCy. J · ∈ ⊕ Thetypenumberof = Φ isaminimaldegreeofthediﬀerentialformΦ,whichisallowedtochange J J on a manifold. 3 Example 2.1. Let J be the ordinary complex structure on a complex manifold X. Then the complex structure J∗ on T∗X is given by (J∗η)(v) := η(Jv) for η T∗X,v TX and we obtain a ∈ ∈ generalized complex structure which is deﬁned by J J J 0 = , J J 0 J∗! − wherethecanonicallinebundleof istheordinarycanonicallinebundleK = n,0 whichconsists J J J ∧ of n-forms. Thuswe haveType =n. J J Example2.2. Letωbearealsymplecticstructureona2n-manifoldM. Thentheinteriorproduct ivω of a vector v yields an isomorphism ω˜ : TM T∗M which admits the inverse ω˜−1. Then a → generalized complex structure is deﬁned by ω J 0 ω˜−1 = − , ψ J ω˜ 0 ! Then thecanonical line bundleof is generated by ω J 1 1 ψ=e√−1ω =1+√ 1ω+ (√ 1ω)2+ + (√ 1ω)n, − 2! − ··· n! − where theminimal degree of ψ is 0. Thuswe haveType =0 ψ J Example 2.3 (Thecationofb-ﬁelds). Let beageneralizedcomplexstructurewhichisinduced J from a non-degenerate, pure spinor φ. A d-closed real 2-form b acts on φ by eb φ which is also a · non-degenerate,purespinor. Thusebφinducesageneralizedcomplexstructure ,whichiscalledthe b J action of b-ﬁeldon . Thegeneralized complex structure b givesthedecomposition TM T∗M = J J ⊕ LJb ⊕LJb, where LJb =Adeb ◦J ◦Ade−b and 1 0 Ad = eb b 1.! Example2.4(Poissondeformations). LetX beacomplexmanifoldandβaholomorphicPoisson structureonacomplexmanifoldX. Thenβgivesdeformationsofnewgeneralizedcomplexstructures byJβt :=Adeβt◦JJ ◦Ade−β where 1 β Ad = eβ 0 1! Thetypenumberis given byType =n 2 rank of β at x M. Jβtx − x ∈ 3 Deformation theory of generalized complex structures Let (M, ) be a generalized complex manifold with the decomposition (TM T∗M)C =L L . J ⊕ J ⊕ J ThebundleL is a Lie algebroid bundlewhich yields the Lie algebroid complex, J 0 0L dL 1L dL 2L dL 3L →∧ J →∧ J →∧ J →∧ J →··· ItisknownthattheLiealgebroid complexisthedeformation complex ofgeneralized complexstruc- tures. In fact, ε 2L gives a deformed isotropic subbundle L := E+[ε,E] E L which ε yields a decompos∈iti∧on (JTM T∗M)C = Lε Lε if ε is suﬃciently sm{all. The iso|trop∈ic bJu}ndle Lε ⊕ ⊕ yieldsageneralizedcomplexstructureifandonlyifεsatisﬁesthegeneralizedMauer-Cartanequation 1 d ε+ [ε,ε] =0, L 2 Sch where[ε,ε]SchdenotestheSchoutenbracket. TheLiealgebroidcomplex( •,dL)isanellipticcomplex ∧ andthespacesofsemi-universaldeformations(theKuranishispaces)ofgeneralizedcomplexstructures are constructed [Gu1]. The space of inﬁnitesimal deformations of generalized complex structure is givenbythesecondcohomology groupH2( •L )andtheobstructionspacesofgeneralized complex ∧ J structureis thethird oneH3( •L ). ∧ J 4 Remark 3.1. Let be the generalized complex structure by the action of d-closed b ﬁelds on b J J. Then the isomorphism Adeb : LJ ∼=LJb yields the isomorphism of the Lie algebroid complexes ∧•LJ ∼=∧•LJb. ThuswehaveanisomorphismoftheLiealgebroidcohomologygroupsHk(∧•LJ)∼= Hk(∧•LJb). Let X =(M,J) be a complex manifold and := denotes the generalized complex structure J J J inducedfrom J as inExample2.1. Thenit turnsoutthat H•( •L ) isthehypercohomology group ∧ J of thetrivial complex of sheaves: 0 0 Θ 0 2Θ 0 3Θ 0 , X →O → →∧ →∧ →··· where 0 denotesthe zero map. Thuswe have Hk(∧•LJJ)=⊕p+q=kHp(X,∧qΘ), whereΘdenotesthesheafofholomorphicvectorﬁeldsonXand qΘdenotestheq-thskew-symmetric ∧ tensor of Θ. The inﬁnitesimal deformations is given by H2(∧•LJJ)=H2(X,OX)⊕H1(X,Θ)⊕H0(X,∧2Θ), where H1(X,Θ) is the inﬁnitesimal deformations of ordinary complex structures and H2(X, ) is O given by the action of b-ﬁelds and H0(X, 2Θ) corresponds to deformations given by holomorphic ∧ Poisson structures. A holomorphic 2-vector β H0(X, 2Θ) is a holomorphic Poisson structure if [β,β] = 0, where Sch ∈ ∧ [, ] stands for the Schouten bracket. A holomorphic 2-vector β gives the Poisson bracket of Sch functionsby f,g =β(df dg). ThenthePoissonbracketsatisﬁestheJacobiidentityifandonlyif β { } ∧ β is a Poisson structure. A holomorphic Poisson structure β satisﬁes the generalized Mauer-Cartan equation since dLβ = ∂β = 0. Thus βt = eβt Je−βt gives deformations of generalized complex J J structures, where t denotes the complex parameter of deformations. We denote by L the Lie Jβ algebroid bundle of . A holomorphic Poisson structure β and the Schouten bracket give a map β J δ : pΘ p+1Θ by δ α := [β,α] , where α pΘ. Since the Schouten bracket satisﬁes the β β Sch ∧ → ∧ ∈ ∧ superJacobi identity and [β,α] =[α,β] , we have Sch Sch 1 δ δ (α)=[β,[β,α] ] = [α,[β,β] ] =0. β◦ β Sch Sch 2 Sch Sch Then we obtain thePoisson complex: 0 δβ Θ δβ 2Θ δβ 3Θ X →O → →∧ →∧ →··· where δ f =[β,f] =[df,β] Θ for f and [df,β] is the commutator in the Cliﬀord algebra β Sch X ∈ ∈O which is equal to the coupling between df and β. A holomorphic Poisson structure β deﬁnes a map β˜ from the sheaf of holomorphic 1-forms Ω1 to Θ by [θ,β] for θ Ω1. The map β˜ gives the map ∈ pβ˜:Ωp pΘ by ∧ →∧ β˜(θ1 θp)=β˜(θ1) β˜(θp). ∧···∧ ∧···∧ Proposition 3.2. The map pβ˜ induces a map from the holomorphic de Rham complex (Ω•,d) ∧ to the Poisson complex ( •Θ,δβ), ∧ 0 // d //Ω1 d //Ω2 d //Ω3 d // X O ··· id β˜ 2β˜ 3β˜ (cid:15)(cid:15) (cid:15)(cid:15) ∧ (cid:15)(cid:15) ∧ (cid:15)(cid:15) 0 // δβ //Θ δβ // 2Θ δβ // 3Θ δβ // X O ∧ ∧ ··· 5 Proof. It follows from our deﬁnitions that β˜(df) = [df,β] = [β,f] = δ f and δ δ (f) = Sch β β β ◦ δ (β˜(df))=0 for f . Sinceδ is a derivation, we have β X β ∈O p+1β˜ d(fdg1 dgp)=β˜(df) β˜(dg1) β˜(dgp) (3.1) ∧ ◦ ∧···∧ ∧ ∧···∧ =δβ(fβ˜(dg1) β˜(dgp)) (3.2) ∧···∧ =δβ pβ˜(fdg1 dgp), (3.3) ◦∧ ∧···∧ where g1, ,gp X. Thus we obtain p+1β˜(dγ)=δβ pβ˜(γ) for γ pΘ. ··· ∈O ∧ ◦∧ ∈∧ The following is already obtained in [LSX] Proposition 3.3. [LSX] The hypercohomology of the Poisson complex is isomorphic to the coho- mology groups of the Lie algebroid complex of , i.e., β J Hk(∧•Θ)∼=Hk(∧•LJβ) for all k. Proof. For the completeness of the paper, we shall give a proof of Proposition 3.3. Let 0,1 be ∧β a vectorbundlewhich is thetwisted 0,1 by theadjoint action of β, ∧ ∧0β,1 :={eβ·θ·e−β =θ+[β,θ]∈∧1,0⊕T0,1M|θ∈∧0,1}. We denote by 0,q the q-th skew symmetric tensor of 0,1. By the twisted Dolbeault operator ∧β ∧β ∂β :=eβ ◦∂◦e−β, we obtain the twisted Dolbeault complex: (∧0β,•,∂β). Let T1,0X be the tangent bundleonXandTp,0Xtheskew-symmetrictensorofT1,0X. Thenwealsohavethecomplex(Tp,0X ⊗ ∧β0,•,∂β) which are a resolution of ∧pΘ. As in the Poisson complex, the map δβ deﬁnes the map between complexes δβ :(Tp,0X⊗∧0β,•,∂β) to(Tp+1,0X⊗∧0β,•,∂β). Thus we haveadouble complex (T•,0X 0,•,δβ,∂β). ItturnsoutthatthetotalcomplexofthedoublecomplexistheLiealgebroid ⊗∧ complex (∧•,LJβ,dL). Since the cohomology groups of the total complex is the hypercohomology groups of thePoisson complex, we haveHk(∧•Θ)∼=Hk(∧•LJβ). 4 Log symplectic structure ω in holomorphic category C and generalized complex structure J φ LetX beacomplexmanifoldofcomplexdimensionn. LetDbeasmoothdivisoronX whichadmits holomorphic coordinates (z1, ,z2m) such that D is given by z1 = 0. We call such coordinates ··· (z1, ,z2n) logarithmic coordinates. A logarithmic symplectic structure† ωC is a d-closed, logarith- ··· mic 2-form on X which satisﬁes the followings (1) and (2) : (1) There exist logarithmic coordinates (z1, ,z2m) on a neighborhood of every point in D such ··· that ωC is written as ωC = dzz11 ∧dz2+dz3∧dz4+···+dz2m−1∧dz2m. (4.1) (2) On the complement X D, ωC is a holomorphic symplectic form. \ Then φ := eωC is a d-closed, non-degenerate, pure spinor on X D which induces the generalized \ complexstructure φ onX D. Onaneighborhood ofD,z1eωC isalsoanondegenerate,purespinor. J \ Hence canbeextendedasageneralizedcomplexstructureonX. Thetypenumberof isgiven φ φ J J bythe followings: On thedivisor D= z1 =0 , φ is induced from z1φ,where { } J z1eωC|z1=0 =dz1∧dz2+···=(dz1∧dz2)∧eω˜C, †Thenotionoflogarithmicsymplecticstructureswasintroducedin[Go3]. 6 whereω˜C =dz3 dz4+ +dz2m 1 dz2m.Thustheminimaldegreeof z1φon D iseuqalto2. On ∧ ··· − ∧ thecomplement X D, is given by φ whose theminimal degree is 0. Thus we have φ \ J 2 (x D) Type (x)= ∈ Jφ 0 (x / D) ∈ 5 Lie algebroid cohomologygroups of J and logarithmic φ Poisson structure β Let ωC be a logarithmic symplectic structure on a complex manifold X. Then ωC gives the isomor- phism between the sheaf of holomorphic logarithmic tangent vectors Θ( logD) and the sheaf of − logarithmic 1-forms Ω1(logD) which also indues the isomorphism 2Θ( logD) =Ω2(logD). Then ∧ − ∼ ωC H0(X,Ω2(logD)) admits the dual 2-vector β H0(X, 2Θ( logD)). Since ωC is d-closed, β ∈ ∈ ∧ − is a Poisson structure. Wecall theβ a holomorphic log Poisson structure. TheinteriorproductivωC byaholomorphicvectorﬁeldv ofωC givesameromorphic1-form with simple pole along D. Thus ωC yields a map ω˜C from Θ to Ωˆ1, where Ωˆ1 is deﬁned to be the image of ω˜C. Then the map ω˜C is the inverse of the map β˜ : Ωˆ1 Θ as in Section 3. We deﬁne Ωˆp by → Ωˆp = pΩˆ1. Thentheexteriorderivativedgivesacomplex(Ωˆ•,d). Since pωˆC givesanisomorphism ∧ ∧ ∧pΘ ∼= Ωˆp which is the inverse of ∧pβ˜, it follows from Proposition 3.2 that the complex (∧•Ωˆ,d) is isomorphic to the Poisson complex (∧•Θ,δβ) and we obtain Hk(∧•Θ) ∼= Hk(Ωˆ•). We denote by (Ω•(logD),d) the holomorphic log complex. Sincethelogcomplex(Ω•(logD),d)isasubcomplexof(Ωˆ•,d),wehavetheshortexactsequence of complexes: 0 Ω•(logD) Ωˆ• Q• 0, → → → → where Q• denotes thequotient complex. Lemma 5.1. Let •(Q•) be the cohomology sheaves of the complex Q•. Then we have that H •(Q•)= 0 . H { } Proof. Let (z1, ,z2n) belogarithmic coordinates of a neighborhood of x D such thatωC is ··· ∈ given by (4.1). Then we see that i∂∂z1ωC = dzz12, i∂∂z2ωC =−dzz11 It follows that the germ of theimage Ωˆ1 is generated by x hdzz12,dzz11,dz3,··· ,dz2mi over . Then we see that thegerm of Ωˆ2 is generated by OX,x x dz1 dz2,dz1 dz ,dz2 dz ,dz dz i,j =1, 2m h z1 ∧ z1 z1 ∧ i z1 ∧ j i∧ j| ··· i over . Then it turnsout that every α Ωˆ2 is written as OX,x ∈ x α=α0 dz1 dz2 +α1 dz1 +α2 dz2 +α3 ∧ z1 ∧ z1 ∧ z1 ∧ z1 where α0,α1,α2,α3 are holomorphic forms. Since d(dz2)= dz1 dz2, it follows that z1 − z1 ∧ z1 γ :=α+(−1)|α0|d(α0∧ dzz12)∈Ωˆ2x is a meromorphic form with pole of order at most one on D. We assume that dα is a logarithmic form. Then z1dα is holomorphic and z1dγ is also holomorphic. Since z1γ is holomorphic, it follow 7 thatγ isalogarithmicform. Ifαisarepresentativeofthegermofthecohomologysheaves •(Q•)x, H then[α]=[γ] •(Q•)x anddαisalogarithmicform. Thusitfollowsthat[α]=[γ]=0 •(Q•)x ∈H ∈H since γ is a logarithmic form and γ 0 in Q•. ≡ Proposition5.2. Thecomplex( •Ωˆ,d)isquasi-isomorphictothelogarithmiccomplex(Ω•(logD),d). ∧ Thus the cohomology groups Hk(∧•Θ)∼=Hk(∧•Ωˆ)∼=Hk(∧•Lφ) are given by Hk(X\D,C). Proof. ItfollowsfromLemma5.1thatthemap(Ω•(logD),d) (Ωˆ•,d)isaquasi-isomorphism. → ThuswehaveHk(Ω•(logD))∼=Hk(Ωˆ•). ItisknownthatthehypercohomologygroupsHk(∧•Ω(logD)) ofthelogcomplexarethecohomologygroupsofthecomplementHk(X D,C). ItfollowsfromPropo- \ sition 3.3 that Hk(∧•Lφ)∼=Hk(∧•Θ). Thuswe obtain Hk(∧•Lφ)∼=Hk(X\D,C). 6 Unobstructed deformations of generalized complex struc- tures induced from C∞ logarithmic symplectic structures Let M be a diﬀerentiable manifold of real dimension 4m and D a submanifold of real codimension 2. We assume that there is an open cover M = U such that each U is an open set of C2m α α α ∪ with complex coordinates (z1, ,z2m) and D is locally given by z1 = 0 for U D = . We α ··· α { α } α ∩ 6 ∅ say (z1, ,z2m) logarithmic coordinates of D. Note that we do not assume that M is a complex α ··· α manifold. In fact, deﬁningequations of D satisﬁes zα1 =efα,βzβ1 on Uα∩Uβ, where fα,β is a smooth function on U U . α β ∩ Definition 6.1. A C∞ logarithmic symplectic structure‡ ωC isad-closed complex 2-form which satisﬁes the followings: (1) On a neighborhood of D, ωC is locally given by ωC = dzz11 ∧dz2+dz3∧dz4+···+dz2m−1∧dz2m, where (z1, ,z2m) are logarithmic coordinates of D= z1 =0 . ··· { } (2)Onaneighborhood ofthecomplement M D, ωC =b+√ 1ω wherebisad-closed 2-form andω \ − denotesa real symplectic structure. Then φ = eωC is a d-closed, non-degenerate, pure spinor which induces the generalized complex structure φonthecomplementM D. Infact,z1φisanon-degeneratepurespinoronaneighborhood J \ U of D. Thusit follows that φ:=eωC deﬁnesa generalized complex structure on M. φ J Then we havethe following theorem : Theorem6.2. LetωC beaC∞ logarithmicsymplecticstructureonM and φ thegeneralizedcom- J plex structure which is induced from φ=eωC. Then the Lie algebroid cohomology groups Hk( •Lφ) ∧ of is isomorphic to Hk(M D,C). φ J \ Theorem 6.3. Let ωC be a C∞ logarithmic symplectic structure on M and φ the generalized J complex structure whichis induced from φ=eωC. Then deformations of are unobstructed and the φ J space of inﬁnitesimal deformations is given by H2(M D,C). \ In order to prove our theorems, we shall introduce C∞ logarithmic deformations of φ and the J C∞ logarithmic deformations are unobstructed in nextSection. ‡NotethatthenotionofC∞logarithmicsymplecticstructuresisdiﬀerentfromtheoneofsingularsymplecticstructures asin[GMP,GuLi]whosesingularlociarerealcodimension1. 8 7 Logarithmic deformations of J φ AC∞ logarithmicvectorﬁeldV onamanifoldM alongDisaC∞ vectorﬁeldwhichislocallygiven by 2m ∂ ∂ ∂ ∂ V =f1z1∂z1 +g1∂z1 + fi∂zi +gi∂zi, i=2 X where fi,gi(i=1, ,2m) are C∞ functions and (z1, ,z2m) are logarithmic coordinates of D = ··· ··· z1 = 0 . Thus a C∞ logarithmic vector ﬁeld V preserves the ideal (z1) which is an analog of { } the notion of logarithmic vector ﬁelds in complex geometry. We denote by logM the sheaf of C∞ T logarithmic vector ﬁelds. The sheaf logM is locally free which gives a C∞ vector bundle TlogM. T Our generalized complex structure φ gives the decomposition TM T∗M = Lφ Lφ and the Lie J ⊕ ⊕ log algebroid complex (∧•Lφ,dLφ). We deﬁnea subbundleLφ by log C Lφ =Lφ∩(TlogM ⊕T∗M) log Thenweobtainthesubcomplex(∧•Lφ ,dL)oftheLiealgebroid complex(∧•Lφ,dL). Thenwehave Proposition7.1. ThecohomologygroupHk(∧•Llφog)ofthesubcomplex(∧•Llφog,dL)isisomorphic to the cohomology group Hk(M D,C) \ Proof. Let Llφog be the sheaf of C∞ sections of the bundle Llφog and ∧pLlφog the p-th skew sym- log metric tensor of . Then we havethecomplex of sheaves: Lφ 0 0 log dL 1 log dL 2 log dL . (7.1) −→∧ Lφ −→∧ Lφ −→∧ Lφ −→··· Since ∧pLlφog is a soft sheaf, the hypercohomology groups Hk(∧•Llφog) of the complex of sheaves (∧•Llφog,dL)isgivenbyglobalsectionsandwehaveHk(∧•Llφog)∼=Hk(∧•Llφog). Theinteriorproduct ivωC of a vector v by ωC restricted to the complement M D gives a map from TlogM to 1-forms on \ log M\D which inducesamapω˜C from Lφ tothesheafofdiﬀerential1-forms onM\D byω˜C(v+θ)= −ivωC+θ. Then we havethemap ∧pω˜C :∧pLlφog →Ap(M\D) by ∧pω˜C(v1∧···∧vs∧θ1∧···∧θs)=(−1)siv1ωC∧···∧ivsωC∧θ1∧···∧θs, (7.2) where s+t = p and p(M D) denotes the sheaf of p-forms on M D. The map pω˜C gives the A \ \ ∧ log map ∧•ω˜C from the complex (∧•Lφ ,dL) to the de Rham complex (A•(M\D),d). We shall show log thatthemap∧•ω˜C :(∧•Lφ ,dL)→(A•(M\D),d)isquasi-isomorphic. Inordertoobtainthequasi- isomorphism,weshalldeterminethecohomologysheavesHk(∧•Llφog)(U)ofthecomplex(∧•Llφog,dL) restricted to a neighborhood U in the followings two cases : (1)IfU isaneighborhoodofDwithlogarithmiccoordinates(z1,, ,z2m),thenthelogarithmiccoor- ··· dinatesdeﬁnethecomplexstructureonU suchthatωC U isalogarithmicsymplecticstructurewhich | is the dual of holomorphic logarithmic Poisson structure β as in Section 5. Then it turns out that the cohomology groups Hk(∧•Llφog)(U) is given by the hypercohomology groups Hk(∧•(Θ(−logD)) of thePoisson complex of multi-logarithmic tangent vectors: 0 0Θ( logD) δβ 1Θ( logD) δβ 2Θ( logD) δβ →∧ − −→∧ − −→∧ − −→··· The map •ω˜C restricted to U gives an isomorphism from the complex of logarithmic multi-tangent ∧ vectors ( •Θ( logD),δβ) to the complex of logarithmic forms (Ω•(logD),d) which induces the ∧ − isomorphism between cohomology groups Hk(Ω•(logD)) ∼= Hk(∧•Θ(−logD)). It is known that the hypercohomology groups Hk(Ω•(logD)) of the complex of logarithmic forms is Hk(U U D,C). \ ∩ Thuswe haveHk(∧•Llφog)(U)∼=Hk(U\U ∩D,C). (2)IfU isaneighborhoodofthecomplementM D,thenωC U isgivenbyb+√ 1ω. Itfollows that \ | − themap ∧•ω˜C :(∧•Llφog(U),dL)→(A•(U),d) is an isomorphism and Hk(∧•Llφog)(U)∼=Hk(U,C). log Thenitfollowsthatthemap∧•ω˜C :(∧•Lφ ,dL)→(A•(M\D),d)isaquasi-isomorphism. Hence we obtain Hk(∧•Llφog)∼=Hk(M\D,C). 9 Proposition 7.2. The second cohomology group H2(∧•Llφog) ∼= H2(M\D,C) gives unobstructed deformations of generalized complex structures. Proof. AC∞ logarithmic1-formθ isaC∞ 1-formonM D whichiswrittenonaneighborhood \ U of D by 2m θ=f1dzz11 +g1dz1+ fidzi+gidzi, i=2 X where fi,gi are C∞ functions and (z1,··· ,z2m) denotes logarithmic coordinates. Let Tlo∗gM be the sheaf of C∞ sections of C∞ logarithmic 1-forms and ∧pTl∗ogM the p-th skew symmetric tensors of Tlo∗gM. Then we havethe complex of C∞ logarithmic forms: 0→∧0Tlo∗gM −d→∧1Tlo∗gM −d→∧2Tlo∗gM −d→··· It turns out that the hypercohomology groups Hk(∧•Tlo∗gM) of the complex of C∞ logarithmic forms is Hk(M D,C). Thus every element of H2(M D,C) admits a d-closed representative α \ \ ∈ C∞(M,∧2Tlo∗gM). Ifαissuﬃcientlysmall, theneωC+α is ad-closed, non-degenerate,purespinoron M DwhichinducesafamilyofdeformationsofgeneralizedcomplexstructuresonM Dparametrized \ \ byanopensetofH2(M D,C). OnaneighborhoodofDwithlogarithmiccoordinates(z1, ,z2m), \ ··· ωC isgivenbyωC = dz1 dz2+ωˆC andαiswrittenasα= dz1 α1+γ,whereωˆC,α1,γ areC∞ 2- z1 ∧ z1 ∧ forms. ThenwehaveωC+α= dz1 (dz2+α1)+ωˆC+γ.Thenz1eωC+α restrictedtoD= z1 =0 z1 ∧ { } is given by z1eωC+α|z1=0=dz1∧(dz2+α1)∧eωˆC+γ Hence z1eωC+α|z1=0 is also a non-degenerate, pure spinor on D for suﬃciently small α. Thus eωC+α gives deformations of generalized complex structures on M which are parametrized by an open set H2(M D,C). \ 8 Proof of main theorems Proof of Theorem 6.2. Let be the sheaf of germs of smooth sections of the bundle L . φ φ L Then theLie algebroid complex gives the complex of sheaves: 0 0 dL 1 dL 2 dL (8.1) φ φ φ →∧ L →∧ L →∧ L →··· Thehypercohomology groups of thecomplex of sheaves are isomorphic to thecohomology groups of the Lie algebroid complex since p are a soft sheaf. We shall apply the similar argument as in φ ∧ L Proposition 7.1 to the complex (8.1). The interior product ivωC of a vector ﬁeld v by ωC restricted to M D gives the map pω˜C : p φ p(M D) as in (7.2) which yields a map •ω˜C form the \ ∧ ∧ L → A \ ∧ complexofsheaves( • φ,dL)tothedeRhamcomplex( •(M D),d). Weshallshowthemap •ω˜C ∧ L A \ ∧ is a quasi-isomorphism. The sheaves •( • ) of cohomology groups of (8.1) are determined by the following two cases H ∧ LJ (1) and (2): (1)IfU isaneighborhoodofDadmittinglogarithmiccoordinatesofD,thenthelogarithmiccoordi- natesdeﬁnethecomplexstructureonU suchthatωC U isalogarithmicsymplecticstructurewhichis | thedualof holomorphic logarithmic Poisson structureβ as in Section 5. It follows from Proposition 5.2 that the cohomology k( • )(U) is isomorphic to Hk(U U D,C). Thus if x M D, then H ∧ LJ \ ∩ ∈ \ •ω˜C induces an isomorphism ∧ Hk(∧•LJ)x ∼= l−im→ Hk(U) x U ∈ (2)IfU isaneighborhoodofthecomplementM D,thenωC =b+√ 1ω givesanisomorphismfrom \ − thecomplex ( • φ(U),dL) to the derham complex ( •(U),d). ∧ L A 10

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