Frobenius Splitting Methods in Geometry and Representation Theory

258 Pages·2005·1.405 MB·English

by  Michel Brion, Shrawan Kumar (auth.)

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
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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