notes on the topology of complex singularities - nicolaescu

《notes on the topology of complex singularities - nicolaescu》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、otesontheTopologyofComplexSingularitiesiviu
.icolaescuDepartmentofathematicsUniversityofotreDameotreDame,
46556USAnicolaescu.1@nd.eduuly15,2001
ntroductionThealgebraicvarietieshaveplayedaveryimportantroleinthedevelopmentofgeometry.Thelinesandtheconicswherethersttobeinvestig

2、atedandmoreover,thestudyofequationsleadsnaturallytoalgebraicgeometry.Thepastcenturyhaswitnessedtheintroductionofnewideasandtechniques,notablyalgebraictopologyandcomplexgeometry.Thesehadadramaticimpactonthedevelopmentofthissubject.Thereareseveralreasonswhichmakealgebraicvarietiessoat

3、tractive.nonehand,itistheirabundanceandthewealthoftechniquesavailabletostudythemand,ontheotherhand,therearetheoftenunexpectedconclusions.Theseconclusionsleadfrequentlytonewresearchquestionsinotherdirections.Thegaugetheoreticrevolutionofthepasttwodecadeshasonlyincreasedtheroleplayed

4、bytheseobjects.orerecently,SimonDonaldsonhasdrawnattentiontoefschetz'oldtechniquesofstudyingalgebraicmanifoldsbyextendingthemtothemuchmoregeneralcontextofsymplecticmanifolds.
havetoadmitthat
wasnotfamiliarwithefschetz'ideasandthisgavemetheimpetustoteachacourseonthissubjectandwrit

5、eupsemi-formalnotes.Thesecondraisond'^etreofthesenotesismypersonalinterestintheisolatedsingularitiesofcomplexsurfaces.ooselyspeaking,efschetzcreatedaholomorphicversionoforsetheorywhenthetraditionalonewasnotevenborn.Heshowedthataholomorphicmapffromacomplex1manifoldtothecomplexpro

6、jectivelinewhichadmitsonlynondegeneratecriticalpointscontainsalargeamountofnontrivialtopologicalinformationabout.Thisinfor-mationcanberecoveredbyunderstandingthebehaviorofthesmoothbersoffastheyapproachasingularone.aturally,onecaninvestigatewhathappenswhenfhasdegeneratepointsaswe

7、lland,unliketherealcase,therearemanymoretoolsatourdisposalwhenapproachingthisissueintheholomorphiccontext.Thisleadstothelocalstudyofisolatedsingularities.Thesenotescoverthematerial
presentedduringthegraduatecourse
taughtattheUniversityofotreDameinthespringof2000.Thiscourseemphasize

8、dtwosubjects,ef-sc