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1、LecturesonthestructureofalgebraicgroupsandgeometricapplicationsMichelBrionPreenaSamuelV.UmaPrefaceThetheoryofalgebraicgroupshaschieflybeendevelopedalongtwodistinctdirections:linear(or,equivalently,affine)algebraicgroups,andabelianvarieties(complete,connectedalgebraicgro
2、ups).ThisismadepossiblebyafundamentaltheoremofChevalley:anycon-nectedalgebraicgroupoveranalgebraicallyclosedfieldisanex-tensionofanabelianvarietybyaconnectedlinearalgebraicgroup,andtheseareunique.Inthesenotes,wefirstexposetheabovetheoremandrelatedstructureresultsaboutc
3、onnectedalgebraicgroupsthatareneitheraffinenorcomplete.Theclassofanti-affinealgebraicgroups(thosehavingonlyconstantglobalregularfunctions)featuresprominentlyinthesedevelopments.Wethenpresentapplicationstosomeques-tionsofalgebraicgeometry:theclassificationofcompletehomoge-
4、neousvarieties,andthestructureofhomogeneous(ortranslation-invariant)vectorbundlesandprincipalbundlesoverabelianvari-eties.WhilethestructuretheoremspresentedatthebeginningofthesenotesgobacktotheworkofBarsotti,ChevalleyandRosen-lichtinthe1950’s,alltheotherresultsarequi
5、terecent;theyaremainlyduetoSanchodeSalas[Sal03,SS09]andthefirst-namedauthor[Bri09,Bri10a,Bri11,Bri12].Wehopethatthepresentexpositionwillstimulatefurtherinterestinthisdomain.InChap-ter1,thereaderwillfindadetailedoverviewofthecontentsofthesubsequentchaptersaswellassomeop
6、enquestions.ThesenotesoriginateinaseriesoflecturesgivenatChennaiMathematicalInstituteinJanuary2011bythefirst-namedau-thor.Hewarmlythanksalltheattendantsofthelectures,espe-vviciallyV.Balaji,D.S.NagarajandC.S.Seshadri,forstimulatingquestionsandcomments;thehospitalityoft
7、heInstituteofMath-ematicalSciences,Chennai,isalsogratefullyacknowledged.ThethreeauthorswishtothankBalajiforhavingpromptedthemtowriteupnotesofthelectures,andencouragedthemalongtheway;thanksarealsoduetoT.Szamuelyforhisveryhelpfulcommentsandsuggestionsonapreliminaryvers
8、ionofthesenotes.ContentsPrefacev1Overview11.1Chevalley’sstructuretheorem............21.2Rosenlichtdecomposition...............41.3C
