quantum cohomology and $s^1$-actions with isolated fixed points

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1、QUANTUMCOHOMOLOGYANDS1-ACTIONSWITHISOLATEDFIXEDPOINTSEDUARDOGONZALEZAbstract.Thispaperstudiessymplecticmanifoldsthatadmitsemi-freecir-cleactionswithisolatedfixedpoints.Weprove,usingresultsontheSeidelelement[4],thatthe(small)quantumcohomologyofa2ndimensiona

2、lmani-foldofthistypeisisomorphictothe(small)quantumcohomologyofaproductofncopiesofP1.ThisgeneralizesaresultduetoTolmanandWitsman[11].1.IntroductionLet(M,ω)bea2ndimensionalcompact,connected,symplecticmanifold,andlet{λ}=λ:S1−→Symp(M,ω)beasymplecticcircleact

3、iononM,thatis,iftXisthevectorfieldgeneratingtheaction,thenLXω=dιXω=0.Recallthatthe1actionissemi-freeifitisfreeonMMS.Thisisequivalenttosaythattheonlyweightsateveryfixedpointare±1.AcircleactionissaidtobeHamiltonianifthereisaC∞functionH:M−→Rsuchthatιω=−dH.Suc

4、hafunctionisXcalledaHamiltonianfortheaction.TolmanandWeitsmanprovedin[11]thatiftheactionissemi-freeandadmitsonlyisolatedfixedpoints,thentheactionmustbeHamiltonianprovidedthatthereisatleastonefixedpoint.Thereisagreatdealofinformationconcerningthetopologyofma

5、nifoldscarryingsuchactions.ThefirstresultinthisdirectionisduetoHattori[2].HeprovesthatthereisanisomorphismfromthecohomologyringH∗(M;Z)tothecohomologyringofaproductofncopiesofP1.Moreover,thisisomorphismpreservesChernclasses.In[11]TolmanandWeitsmangeneralize

6、Hattori’sresulttoequivariantcohomology.Themainresultofthispaperistoextendthisresulttoquantumcohomology.In§3.1weprovethatMisalmostFanomanifold,thereforewecanusepolynomialcoefficientsΛ:=Q[q1,...,qn]forthequantumcohomologyring.Themaintheoremisthefollowing.arXi

7、v:math/0310114v1[math.SG]8Oct2003Theorem1.1.Let(M,ω)bea2n-dimensionalcompactconnectedsymplecticman-ifold.AssumeMadmitsasemi-freecircleactionwithafinitenon-emptysetoffixedpoints.Thenthereisanisomorphismof(small)quantumcohomologyQH∗(M;Λ)∼=QH∗((P1)n;Λ).Notetha

8、twecancomputedirectlythequantumcohomologyofP1×···×P1togetthefollowingresult.Corollary1.2.The(small)quantumcohomologyofMisgivenby∗∗1nQ[x1,...,xn,q1,...,qn]QH(M;Λ)∼=QH((P);Λ)∼=Date:October,2003.partialsuppor