vertex algebras and the cohomology ring structure of hilbert schemes of points on surfaces

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1、VERTEXALGEBRASANDTHECOHOMOLOGYRINGSTRUCTUREOFHILBERTSCHEMESOFPOINTSONSURFACESWei-pingLi1,ZhenboQin2andWeiqiangWang3Abstract.Usingvertexalgebratechniques,wedetermineasetofgeneratorsforthecohomologyringoftheHilbertschemesofpointsonanarbitrarysmoothprojectivesurfaceoverthefieldofcom

2、plexnumbers.1.IntroductionTheHilbertschemeX[n]ofpointsonasmoothprojectivesurfaceXisadesin-gularizationofthen-thsymmetricproductofX(see[Fog]).AnelementξinX[n]isalength-n0-dimensionalclosedsubschemeofX.Recently,therearetwosur-prisingdiscoveries,mainlyduetotheworkofG¨ottsche[Go1],N

3、akajima[Na1,Na2]andGrojnowski[Gro],thatthesumofthecohomologygroupsH=H∗(X[n])nwithQ-coefficientsoftheHilbertschemesX[n]forn≥0haverelationshipswithmodularformsontheonehandandwithrepresentationsofinfinitedimensionalHeisenbergalgebrasontheotherhand(seeaslotheworkofVafaandWitten[V-W]for

4、connectionswithstringtheory).TheseresultshavebeenusedbyLehn[Leh]toinvestigatetherelationbetweentheHeisenbergalgebrastructureandthecupproductstructureofHn.Inparticular,LehnconstructedtheVirasoroalgebrainageometricfashionandstudiedcertaintautologicalsheavesoverX[n].Someotherrecent

5、workonHilbertschemesincludes[dCM,EGL,Go2,Hai,K-T,LQZ,Wan].Inthispaper,byusingvertexalgebratechniques(see[Bor,FLM,Kac])andgeneralizingtheworkofNakajima,GrojnowskiandLehn[Na1,Gro,Na2,Leh],westudythecohomologyringstructureoftheHilbertschemesX[n].WedeterminearXiv:math/0009132v3[math

6、.AG]14Mar2002theringgeneratorsofH∗(X[n])foranarbitrarysmoothprojectivesurfaceXoverthefieldofcomplexnumbers.Inparticular,werecovertheresultofEllingsrudandStrømme[ES2]forX=P2.Moreprecisely,wefindasetof(n·dimH∗(X))generatorsforthecohomologyringHn,andinterprettherelationsamongtheseLge

7、neratorsintermsofcertainoperatorsinEnd(H)whereH=n≥0Hn.OurresultsalsoclearlyindicatethattherearedeepinterplaysbetweenthegeometryofHilbertschemesandvertexalgebrastructureswhichgobeyondtheHeisenbergandVirasoroalgebras.1991MathematicsSubjectClassification.Primary14C05;Secondary17B69.

8、Keywordsandphrases.Hilbertschemes,projectivesur