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The construction problem in Kahler geometry

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44 pages
The construction problem in Kahler geometry Carlos Simpson, CNRS, Laboratoire J.A. Dieudonne Universite de Nice-Sophia Antipolis Parc Valrose, 06100 Nice Dedicated to Alexander Reznikov One of the most surprising things in algebraic geometry is the fact that al- gebraic varieties over the complex numbers benefit from a collection of metric properties which strongly influence their topological and geometric shapes. The existence of a Kahler metric leads to all sorts of Hodge theoretical re- strictions on the homotopy types of algebraic varieties. On the other hand, a sparse collection of examples shows that the remaining liberty is nontrivially large. Paradoxically, with all of this information, the research field remains as wide open as it was many decades ago, because the gap between the known restrictions, and the known examples of what can occur, only seems to grow wider and wider the more closely we look at it. In spite of the differential-geometric nature of the questions and meth- ods, the origins of the situation are very algebraic. We look at subvarieties of projective space over the complex numbers. The main over-arching problem in algebraic geometry is to understand the classification of algebro-geometric objects. The topology of the usual complex-valued points of a variety plays an important role, because the topological type is a locally constant function on any classifying space.

  • geometric shapes

  • hodge

  • mensional complex

  • numbers

  • has recently

  • can occur

  • varieties

  • complex numbers benefit

  • space over


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TheconstructionprobleminKa¨hlergeometryCarlosSimpson,CNRS,LaboratoireJ.A.Dieudonne´Universit´edeNice-SophiaAntipolisParcValrose,06100Nicecarlos@math.unice.frDedicatedtoAlexanderReznikovOneofthemostsurprisingthingsinalgebraicgeometryisthefactthatal-gebraicvarietiesoverthecomplexnumbersbenefitfromacollectionofmetricpropertieswhichstronglyinfluencetheirtopologicalandgeometricshapes.TheexistenceofaKa¨hlermetricleadstoallsortsofHodgetheoreticalre-strictionsonthehomotopytypesofalgebraicvarieties.Ontheotherhand,asparsecollectionofexamplesshowsthattheremaininglibertyisnontriviallylarge.Paradoxically,withallofthisinformation,theresearchfieldremainsaswideopenasitwasmanydecadesago,becausethegapbetweentheknownrestrictions,andtheknownexamplesofwhatcanoccur,onlyseemstogrowwiderandwiderthemorecloselywelookatit.Inspiteofthedifferential-geometricnatureofthequestionsandmeth-ods,theoriginsofthesituationareveryalgebraic.Welookatsubvarietiesofprojectivespaceoverthecomplexnumbers.Themainover-archingprobleminalgebraicgeometryistounderstandtheclassificationofalgebro-geometricobjects.Thetopologyoftheusualcomplex-valuedpointsofavarietyplaysanimportantrole,becausethetopologicaltypeisalocallyconstantfunctiononanyclassifyingspace.Thusthepartitioningoftheclassificationproblemaccordingtotopologicaltypeprovidesacoarse,andmorecalculable,alter-nativetothepartitionbyconnectedcomponents.Furthermore,thetopologyofavarietystronglyinfluencesitsgeometricproperties.Asociologicalobser-vationisthatthequestforunderstandingthetopologyofalgebraicvarietieshasledtoarichsetoftechniqueswhichfoundapplicationselsewhere,eveninphysics.Perhapsawordaboutthechoiceofgroundfieldisappropriate.Onecouldalsolookattheshapesoftherealpointsofrealalgebraicvarieties.However,byWeierstrassapproximation,prettymuchanythingcanariseifyouletthedegreegetbigenough.Thustheclassicalquestioninthiscaseis“whichshapescanoccurforagivendegree?”Thisissomuchmoredifficult1
thatitconstitutesaseparatesubject.1Takingthealgebraicallyclosedfieldofcomplexesmeansthatwedon’tunwittinglyleaveanypointsoutofthepicture,soitseemsreasonabletoregardthisasonepossiblecanonicalchoice.Historicallyatleast,andprobablyalsoforsomephilosophicalreasons,inves-tigationsintothetopologyofcomplexvarietieshaveinturnbeenmirroredinarithmeticalgebraicgeometryoverotherfields.AlotofwhatwesayforcomplexHodgetheorycouldalsoapplytothep-adicversions.Itisastandardnormformathematicalpapers,todiscusspositivecon-tributionstothecollectiveknowledge.Iwouldliketotakeadvantageofthepresentopportunity,tofocusonazonewherewehavelittleornoknowledge,andwhereIhavenothingnewtoreport,inanycase.Iapologizethereforefortheveryvaguenatureofmostofthediscussion.Ihopethattherefer-ences2willgivereadersaplacetolookforwhateverdetailsareavailable.Withourpurposestatedthisway,weshouldnormallytrytocovermostofmodernKa¨hlergeometry.However,thiswouldbeunreasonable,becauseformanypartsofthesubjectIwouldn’thaveanythingfurthertosaythanjustrepeatingwhatwouldbeinthereferences.So,toningdownourambitionsalittlebit,wegettothedescriptionofwhatIreallywouldliketotalkabout,whichistoisolateacertainfamilyofquestionswhichseemimportant,andaboutwhichverylittleseemstobeknown:howtoconstructalgebraic(orKa¨hler)varietieswithinterestingtopologicalbehavior.Togetgoing,recallthatprojectivespaceisendowedwiththeFubini-Studymetric,ahermitianmetricwhichhastheKa¨hlerpropertywhichwenowexplainintwoways.Thefirstwayistonotethatifyouwritethematrixforthehermitianmetricintermsofalocalsystemofholomorphiccoordi-nates,thentransformthetensorproductbetweendzianddzjtoawedgeproduct,yougetanalternatingformω.Itiswell-definedindependantlyofthechoiceofcoordinates.TheKa¨hlerconditionisthatthetwo-formωisclosed,i.e.itgivesasymplecticstructure.Theseconddefinitionistosay1Wecanhope,however,thattheintuitiondeveloppedbyworkersinthatfield(doarecursivesearchonthereferencesin[28])couldbeofhelpinthecomplexcasewearediscussinghere.2Incompilingthereferences,theonlineversionofAMSMathReviewswasagreathelpinremembering,finding,organizingandwritingthemup(oftenbycut-and-paste).Read-ershavingaccesstothisservicecangreatlyexpandtheamountofreferentialinformationavailable,bysearchingstartingwiththeauthorscitedinourbibliography.Acrucialrecentimprovementhasbeentheinclusionofforwardsearchingtogetfuturepapersreferingtoagivenpaper.Thisresultsinsuchavastamountofinformationthatevenourratherlongreferencelistcanonlybeconsideredasasampling.2
thatametricisKa¨hlerifateachpointthereexistsaholomorphicsystemofcoordinatesinwhichthematrixforthemetricisequaltotheidentity,uptoanerrortermin|z|2(ratherthanjust|z|aswouldbethecaseforanarbitrarychoiceofcoordinates).Thegoodsystemiscalledanosculatingsystemofcoordinates.ThissecondpointofviewleadsdirectlytothefamousKa¨hleridentities(D0)=1[Λ,D00],(D00)=−−1[Λ,D0]whereD0=andD00=arethe(1,0)and(0,1)-componentsofthedeRhamdifferential,andΛisthepointwiseadjointoftheoperationω.Thecorrespondencebetweenhermitianmetricsand(1,1)-formsiscom-patiblewithrestrictiontosubvarieties,asistheKa¨hlercondition=0,soanysmoothsubvarietyXPninheritsaKa¨hlermetrictoo.Thisextrin-sicdescriptioncanbemodifiedsoastoseemmoreintrinsic,bysayingthattheKa¨hlermetricisobtainedastheChern(1,1)-formofapositivelycurvedmetriconalinebundleLoverX:intheprojectivecaseListhepullbackofthetautologicalbundleOPn(1)onprojectivespace.Inaphilosophicalsense,aKa¨hlermetricshouldbethoughtofascorrespondingtoacurvatureofaconnectiononalinebundle;thiswillonlybepreciseifthecohomologyclass[ω]isintegral.Ontheotherhand,theremaybealgebraicvarieties,orothermoreexoticthingssuchasalgebraicspaces,whichdon’thaveprojec-tiveembeddingsandsoaren’tKa¨hler.Nonetheless,themainintentionofthetheoryistoinvestigatethepropertiesofprojectivevarieties.TherearesomesignificantdifferencesbetweentheKa¨hlerandtheprojectivesituations[194],butinspiteofthosewewilloftenmixthingsup,sothatquestionsposedforoneclassofvarietiesmightwellbereinterpretedforanyotherrelatedclass.WhenavarietyXhasaKa¨hlermetricω,theKa¨hleridentitiesimplythattheLaplaciansfordandD00areproportional.Thisleads3totheHodgedecompositionforcohomology(certainlywhenXiscompact,andsomere-sultsarepossibleinthenoncompactcasetoo[63][156][183][201]).ItgivesusafirstHodge-theoreticresultaboutthehomotopyofX:ifiisoddthenbi(X):=dimHi(X,C)iseven.4JohnsonandReespointoutthatthisresultcarriesovereasilytothecaseofcohomologywithcoeficientsinalocalsystem3Somegoodreferencesforthetheorystartingfromthebeginningare[87],[194],[195],.]72[4Thisapparentlycharacterizesexactlythosealgebraicsurfaceswhichareprojective,aswasstatedinSimanca’sMathReviewMR1723831of[10]butIdidn’tfindtheactualreferenceforit.3
withfinitemonodromygroup[104],andtheyuseittoshowthatcertainfreeproductscannotoccur.ThiswasextendedbyArapura[11]tocovercertainamalgamatedproducts,inamoreelementaryversionofGromov’generaltheoremthatπ1(X)canneverbeanontrivialamalgamatedproduct[88].Treatingthenoncompactcasebycompactifying,resolvingsingularities,5andthenusingthelogarithmicdeRhamcomplex,leadstomixedHodgestructuresonthecohomologyinthiscase[63][183][201][173][172][149].CombiningwithsimplicialtechniquesgivesmixedHodgestructuresonsin-gularvarietiestoo.AclassicaltechniqueforanalyzingthetopologyofalgebraicvarietiesisLefschetzdevissage[63].Onecanarrange(byabirationaltransformation)tohaveafibrationXP1whosefibersaren1-dimensionalvarieties,afinitenumberofthemsingular(thatistosaymoresingularthantheothers).ThetopologicalpropertiesofXaredeterminedbythetopologicalpropertiesofthegeneralfiber,thewayitstopologyistransformedunderthemon-odromyrepresentationofπ1(P1−{s1,...,sk}),andthewaythesingularfibersfillthingsin.ThisledtothingssuchasZariski’scalculationsoffun-damentalgroups,aswellasproofsofcertainpartsoftheLefschetztheorems(fullerandeasierproofsbeingobtainedusingtheHodgedecompositionforcohomology).GriffithsproposedtothrowHodgetheoryintothissituation,namelytolookathowtheHodgedecompositionofthecohomologyofthefibersvaries.HediscoveredthefundamentalGriffithstransversalityrelationgoverningthevarianceoftheHodgefiltration,anddevelopedmuchofthebasictheoryofvariationsofHodgestructurewhichshowshowthisfamilyofHodgestructuresintegratestogivetheHodgestructureonthetotalspace.Thisleftopenthequestionofwhathappensnearthesingularities,thean-swersfurnishedby[174][51].Similarly,whathappenswhenthetotalspace,andhencethefibers,arenoncompactand/orsingular—inthiscaseweobtainvariationsofmixedHodgestructure,andthecombinedsituationwherethisdegeneratesleadstoamountainoflinearalgebra[183][201]whichveryfewcangrasp(anyoneoutthere?).Towrapupthishistoricalperspectivealittlefaster,basicallythere-mainingtheoreticaldirectionshavebeentolookatHodgetheoryforthefullhomotopytypeofX.Thiscanmeanlookingatthehomotopygroups,whichobtainmixedHodgestructures6[141],idemforthepro-nilpotentcompletion5Hironaka’sresult[98]hasrecentlybeenrevisitedbyseveralpeople[1][24][36].6Onenotableexceptiontotheprinciplethathomotopicinvariantswhicharefinitedi-4