TOPOLOGICAL METHDOS IN ALGEBRAIC GEOMETRY 3RD ED

Inrecentyearsnewtopologicalmethods，especiallythetheoryofsheavesfoundedbyJ.LERAY，havebeenappliedsuccessfullytoalgebraicgeometryandtothetheoryoffunctionsofseveralcomplexvariables.H.CARTANandJ.-P.SERREhaveshownhowfundamentaltheoremsonholomorphicallycompletemanifolds（STEINmanifolds）canbeformulatedintermsofsheaftheory.Thesetheoremsimplymanyfactsoffunctiontheorybecausethedomainsofholomorphyareholomorphicallycomplete.Theycanalsobeappliedtoalgebraicgeometrybecausethecomplementofahyperplanesectionofanalgebraicmanifoldisholomorphicallycomplete.J.-P.SERREhasobtainedimportantresultsonalgebraicmanifoldsbytheseandothermethods.Recentlymanyofhisresultshavebeenprovedforalgebraicvarietiesdefinedoverafieldofarbitrarycharacteristic.K.KODAIRAandD.C.SPENCERhavealsoappliedsheaftheorytoalgebraicgeometrywithgreatsuccess.TheirmethodsdifferfromthoseofSERREinthattheyusetechniquesfromdifferentialgeometry（harmonicintegralsetc.）butdonotmakeanyuseofthetheoryofSTEINmanifolds.M.F.ATIYAHandW.V.D.HODGEhavedealtsuccessfullywithproblemsonintegralsofthesecondkindonalgebraicmanifoldswiththehelpofsheaftheory.IwasabletoworktogetherwithK.KODAIRAandD.C.SPENCERduringastayattheInstituteforAdvancedStudyatPrincetonfrom1952to1954.