The equivariant cohomology ring of regular varieti

arXiv:math/0211026v1[math.AG]2Nov2002THEEQUIVARIANTCOHOMOLOGYRINGOFREGULARVARIETIESMICHELBRIONANDJAMESB.CARRELLAbstract.LetBdenotetheuppertriangularsubgroupofSL2(C),TitsdiagonaltorusandUitsunipotentradical.AcomplexprojectivevarietyYendowedwithanalgebraicactionofBsuchthatthefixedpointsetYUisasinglepoint,iscalledregular.AssociatedtoanyregularB–varietyY,thereisaremarkableaffinecurveZYwithaT–actionwhichwasstudiedin[7].Inthisnote,weshowthatthecoordinateringC[ZY]isisomorphicwiththeequivariantcohomologyringH∗T(Y)withcomplexcoefficients,whenYissmoothor,moregenerally,isaB–stablesubvarietyofaregularsmoothB–varietyXsuchthattherestrictionmapfromH∗(X)toH∗(Y)issurjective.Thisisomorphismisobtainedasarefinementofthelocalizationtheoreminequivariantcohomology;itappliese.g.toSchubertvarietiesinflagvarieties,andtothePetersonvarietystudiedin[11].Anotherapplicationofourisomorphismisanaturalalgebraicformulafortheequivariantpushforward.1.PreliminariesLetBbethegroupofuppertriangular2×2complexmatricesofde-terminant1.LetT(resp.U)bethesubgroupofBconsistingofdiago-nal(resp.unipotent)matrices.Wehaveisomorphismsλ:C∗→Tandϕ:C→U,whereλ(t)=t00t−1andϕ(u)=1u01,togethersatisfyingtherelationλ(t)ϕ(u)λ(t−1)=ϕ(t2u).(1)ConsiderthegeneratorsV=˙ϕ(0)=0100andW=˙λ(1)=100−1oftheLiealgebrasLie(U)andLie(T)respectively.Then[W,V]=2V,andAd(ϕ(u))W=W−2uV(2)ThesecondauthorwaspartiallysupportedbytheNaturalSciencesandEngineeringResearchCouncilofCanada.12MICHELBRIONANDJAMESB.CARRELLforallu∈C.Inthisnote,XwilldenoteasmoothcomplexprojectivealgebraicvarietyendowedwithanalgebraicactionofBsuchthatthefixedpointschemeXUconsistsofonepointo.BoththeB–varietyXandtheactionwillbecalledregular.Theno∈XT,sinceXUisT–stable.Moreover,XTisfinitebyLemma1of[7].ThuswemaywriteXT={ζ1=o,ζ2,...,ζr}.(3)Clearlyr=χ(X),theEulercharacteristicofX.LetH∗T(X)denotetheT–equivariantcohomologyringofXwithcomplexcoefficients.Todefineit,letEbeacontractiblespacewithafreeactionofTandletXT=(X×E)/T(quotientbythediagonalT–action).ThenH∗T(X)=H∗(XT).ItiswellknownthattheequivariantcohomologyringH∗T(pt)ofapointisthepolynomialringC[z],wherezdenotesthelinearformontheLiealgebraofTsuchthatz(W)=1.Thedegreeofzis2.ThusH∗T(X)isagradedalgebraoverthepolynomialringC[z]=H∗T(pt)(viatheconstantmapX→pt).Inoursituation,therestrictionmapincohomologyi∗T:H∗T(X)→H∗T(XT),inducedbytheinclusioni:XT֒→X,isinjective[10].By(3),H∗T(XT)=rMj=1H∗T(ζj)∼=rMj=1C[z],soeachα∈H∗T(X)definesanr–tupleofpolynomials(αζ1,...,αζr).Thatis,i∗T(α)=(αζ1,...,αζr).(4)Wewilldefinearefinedversionofthisrestrictionmapin§4below.2.TheB–stablecurvesThroughoutthisnote,acurveinXwillbeapurelyone–dimensionalclosedsubsetofX.OnewhichisstableunderasubgroupGofBiscalledaG–curve.TheB–curvesinXplayacrucialrole,sowewillnextestablishafewoftheirbasicproperties.Proposition1.IfXisaregularB–variety,theneveryirreducibleB–curveCinXhastheformC=B·ζjforsomeindexj≥2.Moreover,everyB–curvecontainso.Inparticular,thereareonlyfinitelymanyirreducibleB–curvesinX,andtheyallmeetato.Proof.Itisclearthatifj≥2,thenC=B·ζjisaB–curveinXcontainingζj,which,bytheBorelFixedPointTheorem,alsocontainso,sinceoistheonlyB–fixedpoint.Conversely,everyB–curveCinXcontainso,andatTHEEQUIVARIANTCOHOMOLOGYRINGOFREGULARVARIETIES3leastoneotherT–fixedpoint.Indeed,sinceohasanaffineopenT–stableneighbourhoodXoinX,thecomplementC−XoisnonemptyandT–stable.ItfollowsimmediatelythatC=B·xforsomex∈XT−o.ConsidertheactionofBontheprojectivelineP1givenbytu0t−1·z=zt(t+uz)(theinverseofthestandardaction).Ithas(P1)T={0,∞}and(P1)U={0}andisthereforeregular.Notethatifu∈C∗,tu0t−1·∞=1tu,soϕ(u)·∞=u−1.ThediagonalactionofBonX×P1isalsoregular.ByProposition1,theirreducibleB–curvesinX×P1areoftheformB·(x,∞)orB·(x,0),wherex∈XT.Onlythefirsttypewillplayarolehere.ThusputZj=B·(ζj,∞),andletπj:Zj→P1bethesecondprojection.Clearlyeveryπjisbijective,henceZj∼=P1.Inaddition,Zj={(ϕ(u)·ζj,u−1)|u∈C∗}∪{(ζj,∞)}∪{(o,0)},soZi∩Zj={(o,0)}aslongasi6=j.Moreover,restrictingπjgivesanisomorphismpj:Zj−(ζj,∞)→A1.Finally,putZ=[1≤j≤rZj.ThusZistheunionofallirreducibleB–stablecurvesinX×P1thataremappedontoP1bythesecondprojectionπ:X×P1→P1.3.ThefundamentalschemeZLetAdenotethevectorfieldonX×A1definedbyA(x,v)=2Vx−vWx.(5)ObviouslyAistangenttothefibresoftheprojectiontoA1.By(2),(Ad(ϕ(u))W)x=−uA(x,u−1).(6)Thecontractionoperatori(A)definesasheafofidealsi(A)(Ω1X×A1)ofthestructuresheafofX×A1.LetZdenotetheassociatedclosedsubschemeofX×A1.Inotherwords,ZisthezeroschemeofA.Remark1.ThevectorfieldAhereisavariantofthevectorfieldstudiedin[7].Bothhavethesamezeroscheme.4MICHELBRIONANDJAMESB.CARRELLThepropertiesofZfigureprominentlyinthisnote.FirstputXo={x∈X|limt→∞λ(t)·x=o}.(7)Clearly,XoisT–stable;anditfollowseasilyfrom(1)thatXoisopeninX(seeProposition1of[7]fordetails).HencebytheBialynicki–Biruladecompositiontheorem[3],XoisT–equivariantlyisomorphictothetangentspaceToX,whereTactsbyitscanonicalrepresentationatafixedpoint.TheweightsoftheassociatedactionofλonToXareallnegative.Sowemaychoosecoordinatesx1,...,xnonXo∼=ToXthatareeigenvectorsofT;theweightaiofxiisapositiveinteger(itturnsouttobeeven,see[2]).ThisidentifiesthepositivelygradedringC[Xo