An introduction on complex algebraic geometry.pdf

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1、ANINTRODUCTIONTOCOMPLEXALGEBRAICGEOMETRYWITHEMPHASISONTHETHEORYOFSURFACESByChrisPetersMathematischInstituutderRijksuniversiteitLeidenandInstitutFourierGrenobleiPrefaceThesenotesarebasedoncoursesgiveninthefallof1992attheUniversityofLeidenandinthespringof1993attheUniversityofGrenoble.Th

2、esecoursesweremeanttoelucidatetheMoripointofviewonclassificationtheoryofalgebraicsurfacesasbrieflyalludedtoin[P].Thematerialpresentedhereconsistsofamoreorlessself-containedadvancedcourseincomplexalgebraicgeometrypresupposingonlysomefamiliaritywiththetheoryofalgebraiccurvesorRiemannsurfa

3、ces.Butthegoal,asinthelectures,istounderstandtheEnriquesclassificationofsurfacesfromthepointofviewofMori-theory.Inmyopininionanyseriousstudentinalgebraicgeometryshouldbeacquaintedassoonaspossiblewiththeyogaofcoherentsheavesandso,afterrecallingthebasicconceptsinalgebraicgeometry,Ihavetr

4、eatedsheavesandtheircohomologytheory.ThispartculminatedinSerre’stheoremsaboutcoherentsheavesonprojectivespace.Havingmasteredthesetools,thestudentcanreallystartwithsurfacetheory,inparticularwithintersectiontheoryofdivisorsonsurfaces.Thetreatmentgivenisalgebraic,buttherelationwiththetop

5、ologicalintersectiontheoryiscommentedonbriefly.AfullerdiscussioncanbefoundinAppendix2.Intersectiontheorythenisappliedimmediatelytorationalsurfaces.AbasictoolfromthemodernpointofviewisMori’srationalitytheorem.ThetreatmentforsurfacesiselementaryandIborroweditfrom[Wi].Thestudentdoesn’tnee

6、dallofthematerialinChapter4tounderstandit,butatsomepoint,itisveryusefultohavetheSteinfactorisationatone’sdisposal.ThisisthemainreasontoinsertChapter4beforethematerialontherationalitytheorem.RightfromthebeginningIhaveadoptedadualpointofview.Acomplexprojectivevarietycanbestudiedbothfrom

7、thecomplex-analyticaswellfromthecommutativealgebrapointofview.Forinstance,Ihavetreatedcoherentsheavesandtheircohomologyfromthealgebraicpointofview,sincethisisthemostelementarywaytodo.Ontheotherhand,sometimesitisusefultobeabletolookatsmallersetsthanjustaffineopensetsandthenthecomplextopo

8、logyi