Convexity Properties of the Moment Mapping Re-exam

arXiv:dg-ga/9408001v25Aug1997CONVEXITYPROPERTIESOFTHEMOMENTMAPPINGRE-EXAMINEDREYERSJAMAARAbstract.ConsideraHamiltonianactionofacompactLiegrouponacom-pactsymplecticmanifold.AtheoremofKirwan’ssaysthattheimageofthemomentummappingintersectsthepositiveWeylchamberinaconvexpoly-tope.IpresentanewproofofKirwan’stheorem,whichgivesexplicitinfor-mationonhowtheverticesofthepolytopecomeaboutandonhowtheshapeofthepolytopenearanypointcanbereadofffrominfinitesimaldataonthemanifold.ItalsoappliestosomeinterestingclassesofnoncompactorsingularHamiltonianspaces,suchascotangentbundlesandcomplexaffinevarieties.Contents1.Introduction12.Preliminaries33.Semistabilityandconvexity64.Affinevarieties105.Steinvarieties206.Hamiltonianactionsandconvexity247.Examples29References351.IntroductionLetKbeacompactLiegroupactingsmoothlyonacompactsymplecticmani-foldMandsupposethereexistsamoment(um)mapfortheaction.Thismaphasahostofinterestingproperties,oneofthemostimportantofwhichisthefactthattheintersectionofitsimagewithanyWeylchamberisaconvexpolytope,referredtoasthemomentumpolytopeofM.Thistheorem,whichisduetoKirwan,hasalonghistory,partofwhichInowsummarize.Kostantprovedaconvexitythe-oremfortorusactionsonconjugacyclassesandflagmanifoldsin[17].Atiyahin[2]andGuilleminandSternbergin[5]dealtwiththecaseofgeneralHamiltoniantorusactions.Intheirpaper,GuilleminandSternbergfurtherprovedaconvex-itytheoremforHamiltonianactionsofarbitrarycompactLiegroupsonintegralK¨ahlermanifolds(orprojectivemanifolds),whichwasalsoprovedbyMumfordin[27].KirwansubsequentlyextendedthisresulttoHamiltonianactionsonarbitraryDate:December1994.RevisedJuly1997.ToappearinAdv.inMath.1991MathematicsSubjectClassification.Primary58F06;Secondary14L30,19L10.Keywordsandphrases.Momentummappings,geometricquantization,geometricinvarianttheory.12REYERSJAMAARcompactsymplecticmanifoldsin[14].Manyusefulrefinementsintheprojective-algebraiccaseweremadelaterbyBrionin[3].See[4],[11]and[23]forotherresultsandmorereferences.See[9],[15]and[20]forsomedevelopmentssubsequenttothepresentpaper.AstrikingdifferencebetweenKirwan’sgeneralconvexitytheoremandtheabelianconvexitytheoremofAtiyah-Guillemin-Sternbergliesinthefactthatthelatteroffersfarmorequantitativeinformationontheshapeofthemomentumpolytope.Forexample,intheabeliancaseoneknowsthattheverticesofthepolytopeareimagesoffixedpointsinM,andthattheshapeofthepolytopenearavertexcanbereadofffromtheisotropyactiononthetangentspaceatacorrespondingfixedpoint.ThisfollowsfromacombinationoftheequivariantDarbouxTheoremandMorsetheoryappliedtothecomponentsofthemomentummap.Thegoalofthispaperistoobtainsuchinformationinthenonabeliancaseaswell.ThemainresultisTheorem6.7,whichisasharpenedversionofKirwan’sconvexitytheorem.GivenapointminMmappingtoapointμinthemomentumpolytope,itprovidesadescriptionoftheshapeofthepolytopenearμintermsoftheactionofthestabilizerofmonpolynomialsonthetangentspaceatm.Italsostatesanecessarycriterionforμtobeavertex,whichgeneralizesthecriterionfortheabeliancasereferredtoabove.Otherresultsincludeconvexitytheoremsforactionsonaffinevarieties,Theorem4.9,andcotangentbundles,Theorem7.6.Theorem4.8describestherelationbetweenthemomentumconeofanaffinevarietyandthemomentumpolytopesofitsprojectiveclosureandthedivisoratinfinity.TheseresultsareinspiredbyBrion’streatmentofKirwan’stheoremforpro-jectivevarieties.ItcameasasurprisetomehowwellBrion’salgebro-geometrictechniquescanbeadaptedtoaC∞settingessentiallywithoutsacrificinganyoftheirpower.ThemainreasonwhythisispossibleisthateverypointinMpossessesaninvariantneighbourhoodthatisisomorphicasaHamiltonianK-manifoldto(agermof)acomplexquasi-projectivevariety.Inthelanguageoftheorbitmethod,themomentumpolytopeofMisthe“classi-cal”analogueofthesetofhighestweightsoftheunitaryirreduciblerepresentationsoccurringinthe“quantization”ofM.Thereisalsoaclassicalanalogueofthespaceofhighest-weightvectors.Thiswillbethesubjectofaforthcomingpaper.Thepaperisorganizedasfollows.Section2isareviewofsomebasicfactscon-cerningrepresentationsandmomentummaps.Section3isareviewoftheconvexitytheoremforcomplexprojectivevarieties,whereIhavepresentedtheargumentinsuchamannerthatitcanbeappliedtononcompactvarieties.InSections4and5IproveconvexitytheoremsforcomplexaffineandSteinvarieties.InSection6Iap-plytheseresultstoprovelocalconvexitypropertiesofarbitrarymomentummaps,whenceIderivethemainresult,Theorem6.7.Thelocaldescriptionofthemo-mentumpolytopegivenbythistheorem,althoughexplicit,isunwieldyinpractice,andoneoftenhastoreverttoadhocmethodstocalculatemomentumpolytopes.InSection7Iillustratethisinanumberofexamples,suchasactionsoncotangentbundlesandprojectivespaces.IthankSheldonXu-DongChang,YaelKarshonandEugeneLermanfortheirhelpandencouragement.IamgratefultoLaurentLaeng,DomingoLunaandtherefereeforcorrectinganumberoferrors.CONVEXITYPROPERTIESOFTHEMOMENTMAPPINGRE-EXAMINED32.PreliminariesInthissectionIintroducenotationandreviewbasicmaterialtobeusedlater.2.1.Groups,representations.ThroughoutthispaperKwillbeacompactcon-nectedLiegroupwithafixedmaximaltorusT.ThecomplexificationofKisdenotedbyGandthecomplexificationof