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BLOCH TYPE CONJECTURES AND AN EXAMPLE OF A THREEFOLD OF GENERAL TYPE

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Niveau: Supérieur, Doctorat, Bac+8
BLOCH-TYPE CONJECTURES AND AN EXAMPLE OF A THREEFOLD OF GENERAL TYPE CHRIS PETERS Abstract. The hypothetical existence of a good theory of mixed motives predicts many deep phenomena related to algebraic cycles. One of these, a generalization of Bloch's conjecture says that “small Hodge diamonds” go with “small Chow groups”. Voisin's method [19] (which produces examples with small Chow groups) is analyzed carefully to widen its applicability. A threefold of general type without 1- and 2-forms is exhibited for which this extension yields Bloch's generalized conjecture. MSC Classification 14C15, 14C30 1. Introduction Very little is known about the Chow groups of algebraic varieties. This is even true for 0-cycles on surfaces. Mumford [12] has shown that if Albanese equivalence and rational equivalence coincide on 0-cycles of degree 0, then there are no holomorphic 2-forms. In this case the group of 0-cycles of degree 0 modulo rational equivalence is representable, i.e. isomorphic to an abelian variety (the Albanese variety in this case). Bloch conjectured the converse and in [2] this is shown for surfaces which are not of general type. Later it was shown for several classes of surfaces of general type without holomorphic 2-forms (see [8], [1], [19]).

  • chow group

  • any correspondence

  • rational equivalence

  • roitman's theorem

  • given any

  • small

  • has degree


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BLOCOHF-TAYPTEHRCEOENFJOELCDTUORFESGEANNEDRAALNTEYXPAEMPLECHRISPETERSAbstract.Thehypotheticalexistenceofagoodtheoryofmixedmotivespredictsmanydeepphenomenarelatedtoalgebraiccycles.Oneofthese,ageneralizationofBloch’sconjecturesaysthat“smallHodgediamonds”gowith“smallChowgroups”.Voisin’smethod[19](whichproducesexampleswithsmallChowgroups)isanalyzedcarefullytowidenitsapplicability.Athreefoldofgeneraltypewithout1-and2-formsisexhibitedforwhichthisextensionyieldsBloch’sgeneralizedconjecture.MSCClassification14C15,14C301.IntroductionVerylittleisknownabouttheChowgroupsofalgebraicvarieties.Thisiseventruefor0-cyclesonsurfaces.Mumford[12]hasshownthatifAlbaneseequivalenceandrationalequivalencecoincideon0-cyclesofdegree0,thentherearenoholomorphic2-forms.Inthiscasethegroupof0-cyclesofdegree0modulorationalequivalenceisrepresentable,i.e.isomorphictoanabelianvariety(theAlbanesevarietyinthiscase).Blochconjecturedtheconverseandin[2]thisisshownforsurfaceswhicharenotofgeneraltype.Lateritwasshownforseveralclassesofsurfacesofgeneraltypewithoutholomorphic2-forms(see[8],[1],[19]).ForhigherdimensionsananalogueofMumford’sresultcanbefoundin[3]and[4].Ananalogueof(aweakerformof)Bloch’sconjecturecanbeLformu-latedintermsofthelevelofanon-zeroHodgestructureH=Hp,q,i.e.thelargestdifference|pq|forwhichHp,q6=0.Inparticular,foraprojectivemanifoldXthecohomologygroupHk(X,C)haslevel<(k2p)ifandonlyifHk,0(X)=∙∙∙=Hkp,k(X)=0.ForinstanceH1haslevel<1ifitvanishesandH2haslevel<1ifH2=H1,1.Forsurfacesthetwocombinedmeanexactlythatq=pg=0.Bloch’sconjectureinthiscasestatesthatfor0-cyclesrationalandhomologicalequivalencecoincide.Laterveer[10,Cor.1.10]andC.Schoen[17]havefoundageneralizationofMumford’stheorem:ifforallcyclesofdi-mensionsrationalandhomologicalequivalencecoincide,theneverycohomologygroupHk(X)haslevel<k2s.Bloch’sgeneralizedcon-jectureistheconverse.Seealso[9],wheresuchconjecturesarededucedDate:February2009.1