Quantum cohomology of complete intersections

Arnaud Beauville

Description

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

Sujets

• Europe
• Beauville
• Dimensión
• Quantum cohomology

Informations

• Publié par: pefav
• Nombre de lectures: 9
• Langue: English

Extrait

Quantumcohomologyofcompleteintersections ArnaudBeauville 1 Introduction ThequantumcohomologyalgebraofaprojectivemanifoldXisthecohomo- logyofXendowedwithadifferentalgebrastructure,whichtakesintoaccountthe geometryofrationalcurvesinX.Thisstructurehasbeenfirstdefinedheuristically bythemathematicalphysicists[V,W];arigorousconstruction(andproofofthe associativity,whichishighlynontrivial)hasbeenachievedrecentlybyRuanand Tian[R-T]. Whencomputede.g.forsurfaces,thequantumcohomologylooksrathercom- plicated[C-M].Theaimofthisnoteistoshowthatthesituationimprovesconsi- derablywhenthedimensionbecomeshighwithrespecttothedegree.Ourmain resultis: Theorem. − LetX ⊂ P n+r beasmoothcompleteintersectionofdegree(d 1 ,...,d r ) P anddimensionn ≥ 3,withn ≥ 2(d i − 1) − 1.Letd=d 1 ...d r andδ= P (d i − 1).ThequantumcohomologyalgebraH ∗ (X,Q)isthealgebragenerated bythehyperplaneclassHandtheprimitivecohomologyH n (X,Q) o ,withthere- lations: H n+1 =d 1 d 1...d r d r H δ H ∙ α=0α ∙ β=(α | β) 1 (H n − d 1 d 1...d r d r H δ − 1 ) d forα,β ∈ H n (X,Q) o . ThemethodappliesmoregenerallytoalargeclassofFanomanifolds(see Proposition1below).Itisactuallyastraightforwardconsequenceofthedefinitions –exceptfortheexactvalueofthecoefficientd 1 d 1...d r d r ,whichrequiressomestan- dardcomputationsinthecohomologyoftheGrassmannian.StillIbelievethatthe simplicityoftheresultisworthnoticing. Astherefereepointedout,wegetactuallymorethananabstractpresentation ofthequantumcohomologyalgebrabygeneratorsandrelations.Thepointisthat thepowersofthegeneratorHhaveasimplegeometricinterpretation:denotingby H p ∈ H 2p (X,Z)theclassofacodimensionplinearsection,weobtainforp ≤ n k − X pk − X p H p =H p +(` i )H p − k ,H p =H p − (` i )H p − k , i=0i=0 1 PartiallysupportedbytheEuropeanHCMproject“AlgebraicGeometryinEurope”(AGE), ContractCHRXCT-940557. 1