Scalar curvature and stability of toric varieties

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1、j.differentialgeometry62(2002)289-349SCALARCURVATUREANDSTABILITYOFTORICVARIETIESS.K.DONALDSONAbstractWedefineastabilityconditionforapolarisedalgebraicvarietyandstateaconjecturerelatingthistotheexistenceofaKahlermetricofconstantscalarcurvature.Themainresultofthepapergoessom

2、ewaytowardsverifyingthisconjectureinthecaseoftoricsurfaces.Weprovethat,underthestabilityhypothesis,theMabuchifunctionalisboundedbelowoninvariantmetrics,andthatminimisingsequenceshaveacertainconvergenceproperty.Inthereversedirection,wegivenewexamplesofpolarisedsurfaceswhic

3、hdonotadmitmetricsofconstantscalarcurvature.Theproofsuseageneralframework,developedbyGuilleminandAbreu,inwhichinvariantKahlermetricscorrespondtoconvexfunctionsonthemomentpolytopeofatoricvariety.1.IntroductionThispaperisasteptowardsthesolutionofthegeneralproblemoffindingcon

4、ditionsunderwhichacomplexprojectivevarietyadmitsaKahlermetricofconstantscalarcurvature.Thepatternofthean-sweroneexpectsisthatthisdifferentialgeometricconditionshouldbeequivalenttosomenotionof“stability”inthesenseofGeometricIn-variantTheory.Thisexpectationisprobablynowanite

5、moffolklore:goingbacktosuggestionsputforwardbyYauinthecaseofKahler-Einsteinmetrics,andthemanyresultsofTianandothersinthiscase;reinforcedbyadetailedformalpicturewhichmakescleartheanalogywiththewell-establishedrelationbetweenthestabilityofvectorbun-dlesandYang-Millsconnecti

6、ons[5].Here,webegintheinvestigationofReceived11/13/2002.289290s.k.donaldsontheprobleminthespecialcaseoftoricvarieties,workingwithinagen-eraldifferential-geometricframeworkdevelopedbyGuillemin[11],[12]andAbreu[1].Whilewearenotabletoachieveadecisiveresult,thegeometryinthisco

7、mparativelysimplesituationissurprisinglyrichanddisplaysveryclearlytheinterplaybetweentheanalysisandthestabilitycondition.Whilethegeneralideathatconstantscalarcurvatureshouldbere-latedtostabilityhasbeenmentionedbyanumberofauthorsapreciseformulationofaconjecture,withtheappr

8、opriatestabilitycondition,hasnotyetappearedintheliterature.Thuswebeginbyaddressingthisquestion,i