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Quantum cohomology of complete intersections

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12 pages
Quantum cohomology of complete intersections Arnaud Beauville 1 Introduction The quantum cohomology algebra of a projective manifold X is the cohomo- logy of X endowed with a different algebra structure, which takes into account the geometry of rational curves in X . This structure has been first defined heuristically by the mathematical physicists [V,W]; a rigorous construction (and proof of the associativity, which is highly non trivial) has been achieved recently by Ruan and Tian [R-T]. When computed e.g. for surfaces, the quantum cohomology looks rather com- plicated [C-M]. The aim of this note is to show that the situation improves consi- derably when the dimension becomes high with respect to the degree. Our main result is: Theorem .? Let X ? Pn+r be a smooth complete intersection of degree (d1, . . . , dr) and dimension n ≥ 3 , with n ≥ 2 ∑ (di ? 1)? 1 . Let d = d1 . . . dr and ? = ∑ (di ? 1) . The quantum cohomology algebra H ?(X,Q) is the algebra generated by the hyperplane class H and the primitive cohomology Hn(X,Q)o , with the re- lations: Hn+1 = dd11 .

  • hp ?

  • dimension

  • quantum cohomology

  • now let

  • fano manifold

  • hn ?

  • general linear

  • smooth compact


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QuantumcohomologyofcompleteintersectionsArnaudBeauville1IntroductionThequantumcohomologyalgebraofaprojectivemanifoldXisthecohomo-logyofXendowedwithadifferentalgebrastructure,whichtakesintoaccountthegeometryofrationalcurvesinX.Thisstructurehasbeenfirstdefinedheuristicallybythemathematicalphysicists[V,W];arigorousconstruction(andproofoftheassociativity,whichishighlynontrivial)hasbeenachievedrecentlybyRuanandTian[R-T].Whencomputede.g.forsurfaces,thequantumcohomologylooksrathercom-plicated[C-M].Theaimofthisnoteistoshowthatthesituationimprovesconsi-derablywhenthedimensionbecomeshighwithrespecttothedegree.Ourmainresultis:Theorem.LetXPn+rbeasmoothcompleteintersectionofdegree(d1,...,dr)Panddimensionn3,withn2(di1)1.Letd=d1...drandδ=P(di1).ThequantumcohomologyalgebraH(X,Q)isthealgebrageneratedbythehyperplaneclassHandtheprimitivecohomologyHn(X,Q)o,withthere-lations:Hn+1=d1d1...drdrHδHα=0αβ=(α|β)1(Hnd1d1...drdrHδ1)dforα,βHn(X,Q)o.ThemethodappliesmoregenerallytoalargeclassofFanomanifolds(seeProposition1below).Itisactuallyastraightforwardconsequenceofthedefinitions–exceptfortheexactvalueofthecoefficientd1d1...drdr,whichrequiressomestan-dardcomputationsinthecohomologyoftheGrassmannian.StillIbelievethatthesimplicityoftheresultisworthnoticing.Astherefereepointedout,wegetactuallymorethananabstractpresentationofthequantumcohomologyalgebrabygeneratorsandrelations.ThepointisthatthepowersofthegeneratorHhaveasimplegeometricinterpretation:denotingbyHpH2p(X,Z)theclassofacodimensionplinearsection,weobtainforpnkXpkXpHp=Hp+(`i)Hpk,Hp=Hp(`i)Hpk,i=0i=01PartiallysupportedbytheEuropeanHCMproject“AlgebraicGeometryinEurope”(AGE),ContractCHRXCT-940557.1