lecture on invriant theory

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1、LECTURESONINVARIANTTHEORYIGORV.DOLGACHEVFebruary12,2003iiPrefaceThisbookisbasedonone-semestergraduatecoursesIgaveatMichiganin1994and1998,andatHarvardin1999.ApartofthebookisborrowedfromanearlierversionofmylecturenoteswhichwerepublishedbytheSeoulNationalUniver-sity[22].Thema

2、inchangesconsistofincludingseveralchaptersonalgebraicinvarianttheory,simplifyingandcorrectingproofs,andaddingmoreexamplesfromclassicalalgebraicgeometry.ThelastLectureof[22]whichcontainssomeapplicationstoconstructionofmodulispaceshasbeenomitted.Thebookislit-erallyintendedto

3、beafirstcourseinthesubjecttomotivateabeginnertostudymore.AneweditionofD.Mumford’sbookGeometricInvariantTheorywithap-pendicesbyJ.FogartyandF.Kirwan[75]aswellasasurveyarticleofV.PopovandE.Vinberg[91]willhelpthereadertonavigateinthisbroadandoldsubjectofmathematics.Mostoftheres

4、ultsandtheirproofsdiscussedinthepresentbookcanbefoundintheliterature.Weincludesomeoftheextensivebibliographyofthesubject(withnoclaimforcompleteness).Themainpurposeofthisbookistogiveashortandself-containedexpositionofthemainideasofthetheory.Thesolenoveltyisincludingmanyexam

5、plesillustratingthedependenceofthequo-tientonalinearizationoftheactionaswellasincludingsomebasicconstructionsintoricgeometryasexamplesoftorusactionsonaffinespace.Wealsogivemanyexamplesrelatedtoclassicalalgebraicgeometry.Eachchapterendswithasetofexercisesandbibliographicalno

6、tes.Weassumeonlyminimalprerequisitesforstudents:abasicknowledgeofalgebraicgeometrycoveredinthefirsttwochap-tersofShafarevich’sbook[104]and/orHartshorne’sbook[46],agoodknowledgeofmultilinearalgebraandsomerudimentsofthetheoryoflinearrepresentationsofgroups.Althoughweoftenuses

7、omeofthetheoryofaffinealgebraicgroups,theknowledgeofthegroupGLnisenoughforourpurpose.Iamgratefultosomeofmystudentsandcolleaguesforcriticalremarksandcatchingnumerousmistakesinmylecturenotes.SpecialthanksgotoAna-MariaCastravet,MihneaPopa,JanisStipinsandIvanArzhantsev.iiiConte

8、ntsPrefaceiiiIntroductionix1Thesymbolicmethod11.1Firstexamples...........................