Frobenius Splitting Methods in Geometry and Representation Theory PDF
Preview Frobenius Splitting Methods in Geometry and Representation Theory
Progress in Mathematics Volume231 SeriesEditors HymanBass JosephOesterle´ AlanWeinstein Michel Brion Shrawan Kumar Frobenius Splitting Methods in Geometry and Representation Theory Birkha¨user Boston • Basel • Berlin MichelBrion ShrawanKumar Universite´Grenoble1–CNRS UniversityofNorthCarolina InstitutFourier DepartmentofMathematics 38402St.-Martind’He`resCedex ChapelHill,NC27599 France U.S.A. AMSSubjectClassifications:13A35,13D02,14C05,14E15,14F10,14F17,14L30,14M05,14M15, 14M17,14M25,16S37,17B10,17B20,17B45,20G05,20G15 LibraryofCongressCataloging-in-PublicationData Brion,Michel,1958- Frobeniussplittingmethodsingeometryandrepresentationtheory/MichelBrion, ShrawanKumar. p.cm.–(Progressinmathematics;v.231) Includesbibliographicalreferencesandindex. ISBN0-8176-4191-2(alk.paper) 1.Algebraicvarieties.2.Frobeniusalgebras.3.Representationsofgroups.I.Kumar,S. (Shrawan),1953-II.Title.III.Progressinmathematics(Boston,Mass.);v.231. QA564.B742004 516’.3’53–dc22 2004047645 ISBN0-8176-4191-2 Printedonacid-freepaper. (cid:1)c2005Birkha¨userBoston Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(Birkha¨userBoston,c/oSpringerScience+BusinessMediaInc.,Rights andPermissions,233SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsincon- nectionwithreviewsorscholarlyanalysis.Useinconnectionwithanyformofinformationstorage andretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynow knownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,evenifthey arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare subjecttoproprietaryrights. PrintedintheUnitedStatesofAmerica. (TXQ/HP) 987654321 SPIN10768278 Birkha¨userisapartofSpringerScience+BusinessMedia www.birkhauser.com Contents Preface vii 1 FrobeniusSplitting: GeneralTheory 1 1.1 Basicdefinitions,properties,andexamples. . . . . . . . . . . . . . . 2 1.2 ConsequencesofFrobeniussplitting . . . . . . . . . . . . . . . . . . 12 1.3 Criteriaforsplitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4 Splittingrelativetoadivisor . . . . . . . . . . . . . . . . . . . . . . 35 1.5 Consequencesofdiagonalsplitting . . . . . . . . . . . . . . . . . . . 41 1.6 Fromcharacteristicptocharacteristic0 . . . . . . . . . . . . . . . . 53 2 FrobeniusSplittingofSchubertVarieties 59 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.2 FrobeniussplittingoftheBSDHvarietiesZ . . . . . . . . . . . . . 64 w 2.3 SomemoresplittingsofG/B andG/B×G/B . . . . . . . . . . . . 72 3 CohomologyandGeometryofSchubertVarieties 83 3.1 CohomologyofSchubertvarieties . . . . . . . . . . . . . . . . . . . 85 3.2 NormalityofSchubertvarieties . . . . . . . . . . . . . . . . . . . . . 91 3.3 Demazurecharacterformula . . . . . . . . . . . . . . . . . . . . . . 95 3.4 Schubertvarietieshaverationalresolutions . . . . . . . . . . . . . . 100 3.5 HomogeneouscoordinateringsofSchubertvarieties areKoszulalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4 CanonicalSplittingandGoodFiltration 109 4.1 Canonicalsplitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.2 Goodfiltrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.3 ProofofthePRVKconjectureanditsrefinement. . . . . . . . . . . . 142 5 CotangentBundlesofFlagVarieties 153 5.1 Splittingofcotangentbundlesofflagvarieties . . . . . . . . . . . . . 155 5.2 Cohomologyvanishingofcotangentbundlesofflagvarieties . . . . . 169 5.3 Geometryofthenilpotentandsubregularcones . . . . . . . . . . . . 178 vi Contents 6 EquivariantEmbeddingsofReductiveGroups 183 6.1 Thewonderfulcompactification . . . . . . . . . . . . . . . . . . . . 184 6.2 Reductiveembeddings . . . . . . . . . . . . . . . . . . . . . . . . . 197 7 HilbertSchemesofPointsonSurfaces 207 7.1 Symmetricproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 7.2 Hilbertschemesofpoints . . . . . . . . . . . . . . . . . . . . . . . . 216 7.3 TheHilbert–Chowmorphism . . . . . . . . . . . . . . . . . . . . . . 220 7.4 Hilbertschemesofpointsonsurfaces . . . . . . . . . . . . . . . . . 224 7.5 SplittingofHilbertschemesofpointsonsurfaces . . . . . . . . . . . 227 Bibliography 231 Index 247 Preface In the 1980s, Mehta and Ramanathan made important breakthroughs in the study of SchubertvarietiesbyintroducingthenotionofaFrobeniussplitvarietyandcompatibly splitsubvarietiesforalgebraicvarietiesinpositivecharacteristics. Thiswasrefinedby RamananandRamanathanviatheirnotionofFrobeniussplittingrelativetoaneffective divisor. EventhoughmostoftheprojectivevarietiesarenotFrobeniussplit,thosewhichare haveremarkablegeometricandcohomologicalproperties,e.g.,allthehighercohomol- ogygroupsofamplelinebundlesarezero. Interestingly,manyvarietieswherealinear algebraic group acts with a dense orbit turn out to be Frobenius split. This includes theflagvarieties,whicharesplitcompatiblywiththeirSchubertsubvarieties,relative toacertainampledivisor;Bott–Samelson–Demazure–Hansenvarieties;theproductof twoflagvarietiesforthesamegroupG,whicharesplitcompatiblywiththeirG-stable closedsubvarieties;cotangentbundlesofflagvarieties;andequivariantembeddingsof anyconnectedreductivegroup,e.g.,toricvarieties. TheFrobeniussplittingoftheabovementionedvarietiesyieldsimportantgeometric results: Schubertvarietieshaverationalsingularities,andtheyareprojectivelynormal and projectively Cohen–Macaulay in the projective embedding given by any ample line bundle (in particular, they are normal and Cohen–Macaulay); the corresponding homogeneous coordinate rings are Koszul algebras; the intersection of any number ofSchubertvarietiesisreduced; thefullandsubregularnilpotentconeshaverational Gorensteinsingularities;theequivariantembeddingsofreductivegroupshaverational singularities. Moreover,theirproofsareshortandelegant. Further remarkable applications of Frobenius splitting concern the representa- tion theory of semisimple groups: the Demazure character formula; a proof of the Parthasarathy–RangoRao–Varadarajan–Kostantconjectureontheexistenceofcertain componentsinthetensorproductoftwodualWeylmodules;theexistenceofgoodfil- trationsforsuchtensorproductsandalsoforthecoordinateringsofsemisimplegroups inpositivecharacteristics,etc. ThetechniqueofFrobeniussplittinghasprovedtobesopowerfulintacklingnu- merousandvariedproblemsinalgebraictransformationgroupsthatithasbecomean indispensabletoolinthefield. Whilemuchoftheresearchhasappearedinjournals, nothing comprehensive exists in book form. This book systematically develops the viii Preface theory from scratch. Its various consequences and applications to problems in alge- braicgrouptheoryhavebeentreatedinfulldetailbringingthereadertotheforefronts of the area. We have included a large number of exercises, many of them covering complementarymaterial. Alsoincludedaresomeopenproblems. Thisbookissuitableformathematiciansandgraduatestudentsinterestedingeo- metricandrepresentation-theoreticaspectsofalgebraicgroupsandtheirflagvarieties. In addition, it is suitable for a slightly advanced graduate course on methods of pos- itivecharacteristicsingeometryandrepresentationtheory. Throughoutthebook, we assumesomefamiliaritywithalgebraicgeometry,specifically,withthecontentsofthe firstthreechaptersofHartshorne’sbook[Har–77]. Inaddition,inChapters2to6we assumefamiliaritywiththestructureofsemisimplealgebraicgroupsasexposedinthe booksofBorel[Bor–91]orSpringer[Spr–98]. Wealsorelyonsomebasicresultsof representationtheoryofalgebraicgroups,forwhichwerefertoJantzen’sbook[Jan– 03]. Wewarnthereaderthatthetextprovidesmuchmoreinformationthanisneeded formostapplications. Thus,oneshouldnothesitatetoskipaheadatwill,tracingback asneeded. Thefirst-namedauthorowesmanythankstoS.Druel,S.Guillermou,S.Inamdar, M. Decauwert, and G. Rémond for very useful discussions and comments on pre- liminary versions of this book. The second-named author expresses his gratitude to A.Ramanathan,V.Mehta,N.LauritzenandJ.F.Thomsenforalltheytaughthimabout Frobeniussplitting. Thesecond-namedauthoralsoacknowledgesthehospitalityofthe NewtonInstitute,Cambridge(England)duringJanuary–June,2001,wherepartofthis bookwaswritten. ThisprojectwaspartiallysupportedbyNSF.WethankJ.F.Thom- sen,W.vanderKallenandtworefereesforpointingoutinaccuraciesandsuggesting variousimprovements;andL.Trimblefortypingmanychaptersofthebook. Wethank Ann Kostant for her personal interest and care in this project and Elizabeth Loew of TEXniquesfortakingcareofthefinalformattingandlayout. MichelBrionandShrawanKumar September2004 NotationalConvention Thoseexerciseswhichareusedintheproofsinthetextappearwithastar. Frobenius Splitting Methods in Geometry and Representation Theory
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