AND ORBIT VARIETIES FOR FINITE ALGEBRAIC TRANSFORM

ATributetoC.S.SeshadriPerspectivesinGeometryandRepresentationTheoryPages346–378,HindustanBookAgency,2003,ISBN81-85931-39-9INVARIANTTENSORFIELDSANDORBITVARIETIESFORFINITEALGEBRAICTRANSFORMATIONGROUPSMarkLosik,PeterW.MichorandVladimirL.PopovToC.S.Seshadriontheoccasionofhis70thbirthdayAbstract.LetXbeasmoothalgebraicvarietyendowedwithanactionofafinitegroupGsuchthatthereexistsageometricquotientπX:X→X/G.WecharacterizerationaltensorfieldsτonX/GsuchthatthepullbackofτisregularonX:thesearepreciselyallτsuchthatdivRX/G(τ)0whereRX/GisthereflectiondivisorofX/GanddivRX/G(τ)istheRX/G-divisorofτ.Wegivesomeapplications,inpar-ticulartoageneralizationofSolomon’stheorem.InthelastsectionweshowthatifVisafinitedimensionalvectorspaceandGafinitesubgroupofGL(V),theneachautomorphismψofV/Gadmitsabiregularliftϕ:V→Vprovidedthatψmapstheregularstratumtoitselfandψ∗(RX/G)=RX/G.1991MathematicsSubjectClassification.14L24,14L30.Keywordsandphrases.Finitegroup,orbit,tensorfield,orbitspace,lifting.M.L.andP.W.M.weresupportedby“FondszurF¨orderungderwis-senschaftlichenForschung,ProjektP14195MAT”.TypesetbyAMS-TEX12M.LOSIK,P.W.MICHOR,V.L.POPOV1.IntroductionLetXbeasmoothalgebraicvarietyendowedwithanac-tionofafinitegroupG.AssumethatthereexistsageometricquotientπX:X→X/G(thisisalwaysthecaseifXisquasi-projective,cf.Subsection2.4).InthispaperwestudytheinterrelationsbetweenrationaltensorfieldsonXandX/G.IfτisarationaltensorfieldonX/G,wedefineaG-invariantrationaltensorfieldπ∗X(τ)onXcalledthepullbackofτ.IfθisaG-invariantrationaltensorfieldonX,wedefinearationaltensorfieldπX∗(θ)onX/Gcalledthepushforwardofθ.WehaveπX∗(π∗X(τ))=τandπ∗X(πX∗(θ))=θ.Giventhis,weconsiderthefollowingproblem:LetτbearationaltensorfieldonX/G.Whenisthepullbackπ∗X(τ)regularonX?TothatendweconsidertheLunastratificationofX/G.Let(X/G)1betheunionofallcodimension1strata.Weshowthat(X/G)1iscontainedinthesmoothlocusofX/G.Let(X/G)1=(X/G)11∪...∪(X/G)d1bethedecompositionintoirreduciblecomponents.Weshowthatforanyl=1,...,dandz∈(X/G)l1,x∈π−1X(z)thestabilizerofxisacyclicgroupwhoseimageundertheslicerepresentationisgeneratedbyapseudo-reflectionoforderrldependingonlyonl.WeencodethisinformationintothereflectiondivisorRX/G:=r1R1+...+rdRd∈Div(X/G)whereRlistheprimedivi-sorwhosesupportistheclosureof(X/G)l1.Further,foranynonzerorationaltensorfieldτonX/GandpositivedivisorB∈Div(X/G)wedefinediv(τ)∈Div(X/G),thedivisorofτ,anddivB(τ)∈Div(X/G),theB-divisorofτ.Locallydiv(τ)istheminimumofdivisorsofthecomponentfunctionsofτintheframingsoftangentandcotangentbundles.ThedivisordivB(τ)isobtainedfromdiv(τ)bymeansofsome“modifica-tionalong”B(seeSubsection3.6).Ourmainresultisthatπ∗X(τ)isregularonXifandonlyifdivRX/G(τ)0.INVARIANTTENSORFIELDS3AsacorollaryweobtainageneralizationofSolomon’sthe-orem[So],andprovethatthepushforwardofaG-invariantregulartensorfieldonXthatisskewsymmetricwithrespecttothecovariantentriesisregularonthesmoothlocusofX/G.AnotherapplicationpertainstothecasewhereXisavectorspaceVwithalinearactionofG:weobtainacharacterizationofG-invariantpolynomialsonV⊕sintermsofrationalmulti-symmetriccovarianttensorfieldsonV/G.InthelastsectionweprovethatanyautomorphismψofthealgebraicvarietyV/Gsuchthatψ((V/G)0)⊆(V/G)0andψ∗(RV/G)=RV/GcanbeliftedtoanautomorphismofthealgebraicvarietyV.Theproofisbasedontherelevantresultintheanalyticsetting,[KLM],sowhatwereallyproveisthatanalyticliftisactuallyalgebraic.Throughoutinthispaperweassumethatthebasefieldkisalgebraicallyclosedofcharacteristic0.Inthelastsectionweusetheresultfrom[KLM]thatisprovedfork=C,sopass-ingtothegeneralcaseiscarriedoutbyLefschetz’sprinciple.HowevernotethatourproofsoftheresultsfromSections2–4couldbeextendedmutatismutandistothecasewhencharkispositiveandsubjecttosomenon-divisibilityandmagnitudeconditions;forthispurposeoneshouldusetherelevantreplace-mentoftheslicetheorem[Lu]provedin[BR].Thispaperisthealgebraicgeometricalcompanionofpaper[KLM]wheresimilarresultswereobtainedforanalyticactionsoffinitegroups.WethankYu.Neretinforhelpfuldiscussions.Thanksarealsoduetotherefereeforsuggestionsandcommentswhichhasledtoeliminatingsomeinaccuraciesandimprovementintheexposition.Notation,terminologyandconventions:•Ox,XandTx,Xarerespectivelythelocalringandtan-gentspaceatapointxofanalgebraicvarietyX.•XsmisthesmoothlocusofX.4M.LOSIK,P.W.MICHOR,V.L.POPOV•Div(X)istheWeildivisorgroupofX.•div(f)isthedivisorofarationalfunctionf.•supp(D)isthesupportofD∈Div(X).•Apositivedivisoriscalledprimeifitnotsumoftwopositivedivisors.•mC,DisthemultiplicityofaprimedivisorCinadivisorD.•dxϕisthedifferentialofamorphismϕatapointx.•G·zandGzarerespectivelytheorbitandstabilizerofapointzofasetZendowedwithanactionofagroupG.•ZS:={z∈Z|g·z=zforallg∈S}.•|S|isthenumberofelementsofafinitesetS.Allgroupactionsconsideredinthispaperarealgebraicac-tionsonalgebraicvarieties.2.Preliminaries2.1.Excellentmorphisms.Recallthatamorphismofalge-braicvarietiesϕ:X→Yiscalled´etaleatapointx∈Xifthehomomorphismoflocalringsϕ∗:Oϕ(x),Y→Ox,Xinducestheisomorphismoftheircompletions.Ifx∈Xsmandϕ(x)∈Ysm,thenϕis´etaleatxiffdxϕ:Tx,X→Tϕ(x),Yisanisomorphism.Ifϕis´etaleateachpointofX,itiscalled´etale.Le