ZEROTH POISSON HOMOLOGY OF SYMMETRIC POWERS OF

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1、ZEROTHPOISSONHOMOLOGYOFSYMMETRICPOWERSOFISOLATEDQUASIHOMOGENEOUSSURFACESINGULARITIESPAVELETINGOFANDTRAVISSCHEDLERAbstract.LetX⊂C3beasurfacewithanisolatedsingularityattheorigin,givenbytheequationQ(x,y,z)=0,whereQisaweighted-homogeneouspolynomial.Inparticula

2、r,thisincludestheKleiniansurfacesX=C2/GforG

3、lG⋊Sn),whereWeylisthe0n02n2nWeylalgebraon2ngenerators.Thatis,theBrylinskispectralsequencedegeneratesinthiscase.Intheellipticcase,thisyieldsthezerothHochschildhomologyofsymmetricpowersoftheellipticalgebraswiththreegeneratorsmodulotheircenter,Aγ,forallbutcou

4、ntablymanyparametersγintheellipticcurve.Contents1.Introduction11.1.Mainresult11.2.HochschildhomologyofdeformationsandAlev’sconjecture31.3.GeneralsymmetricproductsandPoisson-invariantfunctionals51.4.AC∗-equivariantvectorbundleonP152.ProofofTheorem1.4.1783.P

5、roofofTheorem1.1.13whenXisnotoftypeAm−1114.ProofofTheorem1.1.13intheAm−1case135.ProofofTheorems1.2.1and1.2.2andCorollary1.2.3156.Acknowledgements16References16arXiv:0907.1715v1[math.SG]10Jul20091.Introduction1.1.Mainresult.Leta,b,cbepositiveintegers,andequ

6、ipC[x,y,z]withaweightgradinginwhich

7、x

8、=a,

9、y

10、=b,and

11、z

12、=c.Inthispaper,weareinterestedinsurfacesX⊂C3withanisolatedsingularityattheorigin,cutoutbyapolynomialQ(x,y,z)=0,whichisweighted-homogeneousofdegreed.SuchsurfaceswerefirststudiedsystematicallybySaito[Sai87]

13、.Forconvenience,wealsoassumethata≤b≤c.ThesurfaceXisequippedwithastandardPoissonbracket,givenbythebivector∂∂∂(1.1.1)π:=(∧∧)y(dQ),∂x∂y∂zDate:July9,2009.1whereyisthenaturalcontractionoperation,whichinthiscaseproducesabivectorfromatrivectorandaone-form.Theabov

14、ebivectoris,moreover,weight-homogeneousofdegreeκ:=d−(a+b+c),andisaPoissonbivector(i.e.,{π,π}=0,where{,}istheSchouten-Nijenhuisbracket).HenceitproducesaPoissonbracketofdegreeκ.Inparticular,whenκ<0,XhasaKleinia