elliptic genera and applications to mirror symmetry

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1、EllipticGeneraandApplicationstoMirrorSymmetryLevA.BorisovDepartmentofMathematics,ColumbiaUniversity,NewYork,NY10027e-mail:lborisov@math.columbia.eduAnatolyLibgoberDepartmentofMathematics,UniversityofIllinois,Chicago,IL60607e-mail:libgober@math.uic.eduAbstractThepapercontainsaproofthatellipticgenus

2、ofaCalabi-YaumanifoldisaJacobiform,findsinwhichdimensionstheellipticgenusisdeterminedbytheHodgenumbersandshowsthatellipticgeneraofaCalabi-Yauhyper-surfaceinatoricvarietyanditsmirrorcoincideuptosign.TheproofofthemirrorpropertyisbasedontheextensionofellipticgenustoCalabi-Yauhypersurfacesintoricvariet

3、ieswithGorensteinsingularities.1IntroductionOneofthemotivationsforthispaperwasanattempttounderstandthoseinvariantsofCalabi-YaumanifoldswhichvalueonthemirrorX∗isdeterminedbythevalueonoriginalmanifoldX.ExamplesofsuchinvariantsthatattractedthemostattentionaretopologicalEulercharacteristic,Hodgenumber

4、sandvariousd-pointfunctions.Inparticulartherelationbetweenthedifferentkindsofd-pointfunctionsyieldthearXiv:math/9904126v1[math.AG]22Apr1999famouspredictionsforenumerativegeometry(cf.[6]).TheinvariantconsideredinthispaperisellipticgenusforCalabi-Yaumanifolds.Ellipticgenusoforienteddifferentiablemanif

5、oldsfirstappearedintheworksofLandweber-StongandOchanine(cf.[18]andreferencesthere)aspartofattemptstofindgenerasatisfyingcertaintopologicalconditionsandalsoasameanforcon-structingnewgeneralizedcohomologytheories(ellipticcohomology).Ellipticgenusdefinedinsuchwayisacertainhomomorphismfromtheringoforient

6、edcobordisms∗abΩSOintotheringofmodularformsforthegroupΓ0(2)={

7、c≡0mod2}.cdThevalueofellipticgenusatthecuspsofΓ0(2)isequaltoAˆ-genusandthesigna-ture.AtthesametimeWittenproposedadescriptionofellipticgenusasanindexofcertainDirac-likeoperatorontheloopspaceormoregenerallyasthetraceofcertainoperatorass

8、ociatedwithaconformalfieldtheory(cf.[18]).Alsoatthe1sametimeWittenproposedageneralizationofellipticgenusforalmostcomplexmanifoldswithfirstChernclassdivisiblebyN(cf.[18]).Aconstructionofsuchgeneralizationwasalsogive