Pre homogeneous spaces and stratified Kahler spaces
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Pre-homogeneous spaces and stratified Kahler spaces J. Huebschmann USTL, UFR de Mathematiques CNRS-UMR 8524 59655 Villeneuve d'Ascq Cedex, France Bielefeld July 22, 2007 1
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- pre-homogeneous spaces
- homogeneous space
- kahler spaces
- g0-pre-homogeneous structure
- exotic stratified
- stratified kahler space
- lie group
- kahler space establishes
Pre-homogeneousspacesandstratifiedKa¨hlerspacesJ.HuebschmannUSTL,UFRdeMathe´matiquesCNRS-UMR852459655Villeneuved’AscqCe´dex,FranceJohannes.Huebschmann@math.univ-lille1.frBielefeldJuly22,20071
AbstractArepresentationofareductiveLiegroupisapre-homogeneousspaceprovidedithasaZariski-openorbit.AstratifiedKa¨hlerspaceisastratifiedsymplecticspacetogetherwithacomplexan-alyticstructurewhichiscompatiblewiththestratifiedsymplecticstructure;inparticulareachstratumisaKa¨hlermanifoldinanobviousfashion.ThenotionofstratifiedKa¨hlerspaceestablishesanintimaterelationshipbetweennilpotentorbits,singularreduc-tion,invarianttheory,reductivedualpairs,Jordantriplesystems,symmetricdomains,andpre-homogeneousspaces.ThepurposeofthetalkistoillustratethegeometryofstratifiedKa¨hlerspaces.ExamplesofstratifiedKa¨hlerspacesabound.TheclosureofaholomorphicnilpotentorbitcarriesanormalKa¨hlerstructure.SymplecticreductioncarriesaKa¨hlermanifoldtoanormalstrat-ifiedKa¨hlerspaceinsuchawaythatthesheafofgermsofpo-larizedfunctionscoincideswiththeordinarysheafofgermsofholomorphicfunctions.Projectivizationofholomorphicnilpo-tentorbitsyieldsexoticstratifiedKa¨hlerstructuresoncomplexprojectivespacesandoncertaincomplexprojectivevarietiesin-cludingcomplexprojectivequadrics,Scorzavarietiesandtheirsecantvarieties.Physicalexamplesarereducedspacesarisingfromangularmomentum.Theappearanceofsingularitiesinclassicalphasespacesistheruleratherthantheexception.Whilewithincanonicalquantiza-tiontheimplementationofsingularitiesisfarfrombeingclearthepossibleimpactofclassicalphasespacesingularitiesonquantumproblemscanbeexploredviaquantizationonstratifiedKa¨hlerspaces.2
