mirror on abelian

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1、1MirrorsymmetryofAbelianvarietyandMultiThetafunctionsbyKenjiFUKAYA（深谷賢治）DepartmentofMathematics,FacultyofScience,KyotoUniversity,Kitashirakawa,Sakyo-ku,KyotoJapanTableofcontents§0Introduction.§1ModulispacesofLagrangiansubmanifoldsandconstructionofamirrortorus.§2Constructi

2、onofasheaffromanaffineLagrangiansubmanifold.§3SheafcohomologyandFloercohomology1(Constructionofahomomorphism).§4Isogeny.§5SheafcohomologyandFloercohomology2(Proofofisomorphism).§6ExtensionandFloercohomology1(0thcohomology).§7Modulispaceofholomorphicvectorbundlesonamirrort

3、orus.§8NontransversalordisconnectedLagrangiansubmanifolds.¥§9MultiThetaseries1(DefinitionandAformulae.)§10MultiThetaseries2(Calculationofthecoefficients.)§11ExtensionandFloercohomology2(Highercohomology).§12ResolutionandLagrangiansurgery.2§0IntroductionInthispaper,westudy

4、mirrorsymmetryofcomplexandsymplectictoriasanexampleofhomologicalmirrorsymmetryconjectureofKontsevich[24],[25]betweensymplecticandcomplexmanifolds.Wediscussedmirrorsymmetryoftoriin[12]emphasizingits“noncom-mutative”generalization.Inthispaper,weconcentrateonthecaseofacommut

5、ative(usual)torus.OurresultisageneralizationofonebyPolishchukandZaslow[42],[41],whostudiedthecaseofellipticcurve.Themainresultsofthispaperestablishadictionaryofmirrorsymmetrybetweensymplecticgeometryandcomplexgeometryinthecaseoftoriofarbitrarydimension.Wewrotethisdictiona

6、ryintheintroductionof[12].Wepresenttheargumentinawaysothatitsuggestsapossibilityofitsgeneralization.Howevertherearevariousseriousdifficultiesforthegeneral-ization,someofwhichwementioninthispaper.Inthispaper,wewilldefineanewfamilyofthetafunctionsoncomplextori,whichwecallmu

7、ltithetafunction.Itisageneratingfunctionofthenumbersobtainedbycountingholomorphicpolygonsintorianddescribevariousproductstructures(Yoneda,andMasseyYonedaproducts)ofthesheafcohomologygrouponitsmirror.Werecallthatonefamousapplication[7]ofmirrorsymmetryisthatageneratingfunct

8、ionofthenumbercountingrationalcurvesinaCalabi-YaumanifoldisequaltotheYukawacoupling,aproductstru