Hodge cycles on abelian varieties P.Deligne(notesbyJ.S.Milne) July4,2003;June1,2011;October1,2018. Abstract ThisisaTeXedcopyofthearticlepublishedin Deligne,P.;Milne,J.S.;Ogus,A.;Shih,Kuang-yen.Hodgecycles,motives, andShimuravarieties. LNM900,Springer,1982,pp.9–100, somewhat revised and updated. See the endnotes for more details. The original M.1 articleconsistedofthenotes(writtenbyjsm)oftheseminar“Pe´riodesdesInte´grals Abe´liennes” given by P. Deligne at I.H.E.S., 1978-79. This file is available at www. jmilne.org/math/Documents. Contents Introduction. . . . . . . . . . . . . . . . . . . . . . . . 2 1 Reviewofcohomology . . . . . . . . . . . . . . . . . . . . 5 Topologicalmanifolds . . . . . . . . . . . . . . . . . . . . 5 Differentiablemanifolds . . . . . . . . . . . . . . . . . . . 5 Complexmanifolds . . . . . . . . . . . . . . . . . . . . . 7 Completesmoothvarieties . . . . . . . . . . . . . . . . . . 7 Applicationstoperiods. . . . . . . . . . . . . . . . . . . . 11 2 AbsoluteHodgecycles;principleB . . . . . . . . . . . . . . . 15 Definitions(k algebraicallyclosedoffinitetranscendencedegree) . . . . 15 BasicpropertiesofabsoluteHodgecycles . . . . . . . . . . . . . 16 Definitions(arbitraryk) . . . . . . . . . . . . . . . . . . . 19 Statementofthemaintheorem . . . . . . . . . . . . . . . . . 19 PrincipleB . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Mumford-Tategroups;principleA . . . . . . . . . . . . . . . 22 Characterizingsubgroupsbytheirfixedtensors . . . . . . . . . . . 22 Hodgestructures . . . . . . . . . . . . . . . . . . . . . . 23 Mumford-Tategroups . . . . . . . . . . . . . . . . . . . . 24 PrincipleA . . . . . . . . . . . . . . . . . . . . . . . . 26 4 ConstructionofsomeabsoluteHodgecycles . . . . . . . . . . . 28 Hermitianforms . . . . . . . . . . . . . . . . . . . . . . 28 ConditionsforVd H1.A;Q/toconsistofabsoluteHodgecycles . . . . 30 E 5 CompletionoftheproofforabelianvarietiesofCM-type . . . . . . 36 AbelianvarietiesofCM-type. . . . . . . . . . . . . . . . . . 36 ProofofthemaintheoremforabelianvarietiesofCM-type. . . . . . . 37 1 CONTENTS 2 6 Completionoftheproof;consequences . . . . . . . . . . . . . 41 CompletionoftheproofofTheorem2.11 . . . . . . . . . . . . . 41 ConsequencesofTheorem2.11. . . . . . . . . . . . . . . . . 42 7 Algebraicityofvaluesofthe(cid:0)-function . . . . . . . . . . . . . 44 TheFermathypersurface . . . . . . . . . . . . . . . . . . . 45 Calculationofthecohomology . . . . . . . . . . . . . . . . . 46 TheactionofGal.Q=k/onthee´talecohomology . . . . . . . . . . 49 Calculationoftheperiods . . . . . . . . . . . . . . . . . . . 51 Thetheorem. . . . . . . . . . . . . . . . . . . . . . . . 53 Restatementofthetheorem . . . . . . . . . . . . . . . . . . 55 References. . . . . . . . . . . . . . . . . . . . . . . . . . . 57 M Endnotes(byJ.S.Milne) . . . . . . . . . . . . . . . . . . . 60 Introduction LetX beasmoothprojectivevarietyoverC. Hodgeconjecturedthatcertaincohomology classesonX arealgebraic. Themainresultprovedinthesenotesshowsthat,whenX isan abelianvariety,theclassesconsideredbyHodgehavemanyofthepropertiesofalgebraic classes. Inmoredetail,letXanbethecomplexanalyticmanifoldassociatedwithX,andconsider thesingularcohomologygroupsHn.Xan;Q/. ThevarietyXan beingofKa¨hlertype(every projective embedding defines a Ka¨hler structure), its cohomology groups Hn.Xan;C/' Hn.Xan;Q/˝Chavecanonicaldecompositions Hn.Xan;C/D M Hp;q; Hp;q Ddef Hq.Xan;˝p /. Xan pCqDn Thecohomologyclasscl.Z/2H2p.Xan;C/ofanalgebraicsubvarietyZ ofcodimensionp inX isrational(i.e.,itliesinH2p.Xan;Q//andisofbidegree.p;p/(i.e.,itliesinHp;p). TheHodgeconjecturestatesthat,conversely,everyelementof H2p.Xan;Q/\Hp;p isaQ-linearcombinationoftheclassesofalgebraicsubvarieties. Sincetheconjectureis unproven,itisconvenienttocalltheserational.p;p/-classesHodgecyclesonX. NowconsiderasmoothprojectivevarietyX overafieldk thatisofcharacteristiczero, algebraically closed, and small enough to be embeddable in C. The algebraic de Rham cohomologygroupsHn .X=k/havethepropertythat,foranyembedding(cid:27)Wk,!C,there dR arecanonicalisomorphisms Hn .X=k/˝ C'Hn .Xan;C/'Hn.Xan;C/: dR k;(cid:27) dR Itisnaturaltosaythatt 2H2p.X=k/isaHodgecycleonX relativeto(cid:27) ifitsimagein dR H2p.Xan;C/is.2(cid:25)i/p timesaHodgecycleonX˝ C. Theargumentsinthesenotes k;(cid:27) 2p showthat,ifX isanabelianvariety,thenanelementofH .X=k/thatisaHodgecycleon dR X relativetooneembeddingofk intoCisaHodgecyclerelativetoallembeddings;further, foranyembedding,.2(cid:25)i/p timesaHodgecycleinH2p.Xan;C/alwaysliesintheimage CONTENTS 3 ofH2p.X=k/.1 ThusthenotionofaHodgecycleonanabelianvarietyisintrinsictothe dR variety: itisapurelyalgebraicnotion. Inthecasethatk DCthetheoremshowsthatthe imageofaHodgecycleunderanautomorphismofCisagainaHodgecycle;equivalently, the notion of a Hodge cycle on an abelian variety over C does not depend on the map X !SpecC. Ofcourse,allthiswouldbeobviousifonlyoneknewtheHodgeconjecture. Infact,astrongerresultisprovedinwhichaHodgecycleisdefinedtobeanelementof Hn .X/(cid:2)Q Hn.X ;Q /. Asthetitleoftheoriginalseminarsuggests,thestrongerresult dR l et l hasconsequencesforthealgebraicityoftheperiodsofabelianintegrals: briefly,itallows onetoproveallarithmeticpropertiesofabelianperiodsthatwouldfollowfromknowingthe Hodgeconjectureforabelianvarieties. M.2 —————————————————- Inmoredetail,themaintheoremprovedinthesenotesisthateveryHodgecycleonan abelianvariety(incharacteristiczero)isanabsoluteHodgecycle—see(cid:144)2forthedefinitions andTheorem2.11foraprecisestatementoftheresult. Theproofisbasedonthefollowingtwoprinciples. A. Lett ;:::;t beabsoluteHodgecyclesonasmoothprojectivevarietyX andletG be 1 N thelargestalgebraicsubgroupofGL.H(cid:3).X;Q//(cid:2)GL.Q.1//fixingthet ;thenevery i cohomologyclasst onX fixedbyG isanabsoluteHodgecycle(see3.8). B. If.Xs/s2S isanalgebraicfamilyofsmoothprojectivevarietieswithS connectedand smoothand.ts/s2S isafamilyofrationalcycles(i.e.,aglobalsectionof...) such thatt isanabsoluteHodgecycleforones,thent isanabsoluteHodgecycleforall s s s (see2.12,2.15). EveryabelianvarietyAwithaHodgecyclet iscontainedinasmoothalgebraicfamily inwhicht remainsHodgeandwhichcontainsanabelianvarietyofCM-type. Therefore, Principle B shows that it suffices to prove the main theorem for A an abelian variety of CM-type (see (cid:144)6). Fix a CM-field E, which we can assume to be Galois over Q, and let ˙ be a set of representatives for the E-isogeny classes over C of abelian varieties with complexmultiplicationbyE. PrincipleBisusedtoconstructsomeabsoluteHodgeclasses on L A — the principle allows us to replace ˚A by an abelian variety of the form A2˙ A0˝ZOE (see(cid:144)4). LetG (cid:26)GL.˚A2˙H1.A;Q//(cid:2)GL.Q.1//bethesubgroupfixingthe absoluteHodgecyclesjustconstructedplussomeother(obvious)absoluteHodgecycles. ItisshownthatG fixeseveryHodgecycleonA,andPrincipleAthereforecompletesthe proof(see(cid:144)5). OnanalyzingwhichpropertiesofabsoluteHodgecyclesareusedintheaboveproof, one arrives at a slightlystronger result.2 Call arational cohomology classc on asmooth projective complex variety X accessible if it belongs to the smallest family of rational cohomologyclassessuchthat: (a) thecohomologyclassofeveryalgebraiccycleisaccessible; (b) thepull-backbyamapofvarietiesofanaccessibleclassisaccessible; (c) ift ;:::;t 2H(cid:3).X;Q/areaccessible,andifarationalclasst insomeH2p.X;Q/ 1 N isfixedbyanalgebraicsubgroupG ofAut.H(cid:3).X;Q//(automorphismsofH(cid:3).X;Q/ asagradedalgebra)fixingthet ,thent isaccessible; i 1Added(jsm).Thisdoesn’tfollowdirectlyfromTheorem2.11(see2.4).However,oneobtainsavariantof Theorem2.11usingtheabovedefinitionssimplybydroppingthee´talecomponenteverywhereintheproof(see, forexample,2.10b). 2Added(jsm).Seealso(cid:144)9ofMilne,Shimuravarietiesandmoduli,2013. CONTENTS 4 (d) PrincipleB holdswith“absoluteHodge”replacedby“accessible”. Sections4,5,6ofthesenotescanbeinterpretedasprovingthat,whenX isanabelianvariety, everyHodgecycleisaccessible. Sections2,3definethenotionofanabsoluteHodgecycle andshowthatthefamilyofabsoluteHodgecyclessatisfies(a),(b),(c),and(d);therefore, anaccessibleclassisabsolutelyHodge. Wehavetheimplications: M.3 abelianvarieties trivial HodgeHHHHHHHH)accessibleHHHH)absolutelyHodgeHHHH)Hodge. Onlythefirstimplicationisrestrictedtoabelianvarieties. The remaining three sections, (cid:144)1 and (cid:144)7, serve respectively to review the different cohomologytheoriesandtogivesomeapplicationsofthemainresultstothealgebraicityof productsofspecialvaluesofthe(cid:0)-function. NOTATION We define C to be the algebraic closure of Rpand i 2C to be a square root of (cid:0)1; thus i isonlydefineduptosign. Achoiceofi D (cid:0)1determinesanorientationofCasareal manifold—wetakethatforwhich1^i >0—andhenceanorientationofeverycomplex manifold. ComplexconjugationonCisdenotedby(cid:19)orbyz7!z. Recallthatthecategoryofabelianvarietiesuptoisogenyisobtainedfromthecategory of abelian varieties by taking the same class of objects but replacing Hom.A;B/ with Hom.A;B/˝Q. Weshallalwaysregardanabelianvarietyasanobjectinthecategoryof abelianvarietiesuptoisogeny: thusHom.A;B/isavectorspaceoverQ. If.V /isafamilyofrepresentationsofanalgebraicgroupG overafieldkandt 2V , ˛ ˛;ˇ ˛ thenthesubgroupofG fixingthet isthealgebraicsubgroupH ofG suchthat,forall ˛;ˇ k-algebrasR, H.R/Dfg2G.R/jg.t ˝1/Dt ˝1,all˛;ˇg. ˛;ˇ ˛;ˇ _ Lineardualsaredenotedby . IfX isavarietyoverafieldk and(cid:27) isahomomorphism (cid:27)Wk,!k0,then(cid:27)X denotesthevarietyX˝ k0 (DX(cid:2) Spec.k0/). k;(cid:27) Spec.k/ Bya(cid:29)b wemeanthataissufficientlygreaterthanb. 1 REVIEWOFCOHOMOLOGY 5 1 Review of cohomology Topological manifolds LetX beatopologicalmanifoldandF asheafofabeliangroupsonX. Weset Hn.X;F/DHn.(cid:0).X;F(cid:15)// where F !F(cid:15) is any acyclic resolution of F. This defines Hn.X;F/ uniquely up to a uniqueisomorphism. WhenF istheconstantsheafdefinedbyafieldK,thesegroupscanbeidentifiedwith singularcohomologygroupsasfollows. LetS(cid:15).X;K/bethecomplexinwhichSn.X;K/is theK-vectorspacewithbasisthesingularn-simplicesinX andtheboundarymapsendsa simplextothe(usual)alternatingsumofitsfaces. Set S(cid:15).X;K/DHom.S(cid:15).X;K/;K/ withtheboundarymapforwhich .˛;(cid:27)/7!˛.(cid:27)/WS(cid:15).X;K/˝S(cid:15).X;K/!K isamorphismofcomplexes,namely,thatdefinedby .d˛/.(cid:27)/D.(cid:0)1/deg.˛/C1˛.d(cid:27)/: PROPOSITION 1.1. ThereisacanonicalisomorphismHn.S(cid:15).X;K//!Hn.X;K/. PROOF. IfU istheunitball,thenH0.S(cid:15).U;K//DK andHn.S(cid:15).U;K//D0forn>0. Thus,K !S(cid:15).U;K/isaresolutionofthegroupK. LetSn bethesheafofX associated with the presheaf V 7!Sn.V;K/. The last remark shows that K !S(cid:15) is a resolution of thesheafK. AseachSn isfine(Warner1971,5.32),Hn.X;K/'Hn.(cid:0).X;S(cid:15)//. Butthe obviousmapS(cid:15).X;K/!(cid:0).X;S(cid:15)/issurjectivewithanexactcomplexaskernel(loc. cit.), andso Hn.S(cid:15).X;K//!' Hn.(cid:0).X;S(cid:15)//'Hn.X;K/. (cid:4) Differentiable manifolds NowassumeX isadifferentiablemanifold. Onreplacing“singularn-simplex”by“differen- 1 tiablesingularn-simplex”intheabovedefinitions,oneobtainscomplexesS .X;K/and (cid:15) (cid:15) S .X;K/. Thesameargumentshowsthatthereisacanonicalisomorphism 1 Hn.X;K/Ddef Hn.S1.X;K//!' Hn.X;K/ 1 (cid:15) (Warner1971,5.32). LetOX1 bethesheafofC1real-valuedfunctionsonX,let˝Xn1 betheOX1-module 1 (cid:15) ofC differentialn-formsonX,andlet˝ bethecomplex X1 OX1 !d ˝X11 !d ˝X21 !d (cid:1)(cid:1)(cid:1): 1 REVIEWOFCOHOMOLOGY 6 ThedeRhamcohomologygroupsofX aredefinedtobe fclosedn-formsg Hn .X/DHn.(cid:0).X;˝(cid:15) //D : dR X1 fexactn-formsg IfU istheunitball,Poincare´’slemmashowsthatH0 .U/DRandHn .U/D0forn>0. dR dR Thus,R!˝(cid:15) isaresolutionoftheconstantsheafR,andasthesheaves˝n arefine X1 X1 (Warner1971,5.28),wehaveHn.X;R/'Hn .X/. dR For! 2(cid:0).X;˝n /and(cid:27) 2S1.X;R/,define X1 n Z h!;(cid:27)iD.(cid:0)1/n.n2C1/ ! 2R. (cid:27) Stokes’stheoremstatesthatR d! DR !,andso (cid:27) d(cid:27) hd!;(cid:27)iC.(cid:0)1/nh!;d(cid:27)iD0. Thepairingh;ithereforedefinesamapofcomplexes fW(cid:0).X;˝(cid:15) /!S(cid:15) .X;R/. X1 1 THEOREM 1.2 (DE RHAM). The map HdnR.X/!H1n.X;R/ defined by f is an isomor- phismforalln. PROOF. Themapisinversetothemap ' Hn.X;R/!Hn.X;R/'Hn .X/ 1 dR definedintheprevioustwoparagraphs(Warner1971,5.36). (Oursignsdifferfromtheusual signsbecausethestandardsignconventions Z Z Z Z Z d! D !; pr(cid:3)!^pr(cid:3)(cid:17)D !(cid:1) (cid:17); etc. 1 2 (cid:27) d(cid:27) X(cid:2)Y X Y violatethesignconventionsforcomplexes.) (cid:4) A number R !, (cid:27) 2H .X;Q/, is called a period of !. The map in (1.2) identifies (cid:27) n Hn.X;Q/withthespaceofclassesofclosedformswhoseperiodsareallrational. Theorem 1.2canberestatedasfollows: acloseddifferentialformisexactifallitsperiodsarezero; thereexistsacloseddifferentialformhavingarbitrarilyassignedperiodsonanindependent setofcycles. REMARK 1.3 (SINGER AND THORPE 1967, 6.2). If X is compact, then it has a smooth (cid:15) triangulationT. DefineS(cid:15).X;T;K/andS .X;T;K/asbefore,butusingonlysimplicesin T. Thenthemap (cid:0).X;˝(cid:15) /!S(cid:15).X;T;K/ X1 definedbythesameformulasasf aboveinducesisomorphisms Hn .X/!Hn.S(cid:15).X;T;K//. dR 1 REVIEWOFCOHOMOLOGY 7 Complex manifolds (cid:15) NowletX beacomplexmanifold,andlet˝ denotethecomplex Xan O !d ˝1 !d ˝2 !d (cid:1)(cid:1)(cid:1) Xan Xan Xan inwhich˝n isthesheafofholomorphicdifferentialn-forms. Thus,locallyasectionof Xan ˝n isoftheform Xan X ! D ˛ dz ^:::^dz i1:::in i1 in with ˛ a holomorphic function and the z complex local coordinates. The complex i1:::in i formofPoincare´’slemmashowsthatC!˝(cid:15) isaresolutionoftheconstantsheafC,and Xan sothereisacanonicalisomorphism Hn.X;C/!Hn.X;˝(cid:15) / (hypercohomology). Xan IfX isacompactKa¨hlermanifold,thenthespectralsequence Ep;q DHq.X;˝p / H) HpCq.X;˝(cid:15) / 1 Xan Xan degenerates,andsoprovidesacanonicalsplitting M.4 Hn.X;C/D M Hq.X;˝p / (theHodgedecomposition) Xan pCqDn as Hp;q Ddef Hq.X;˝p / is the complex conjugate of Hq;p relative to the real structure Xan Hn.X;R/˝C'Hn.X;C/ (Weil 1958). The decomposition has the following explicit description: the complex ˝(cid:15) ˝C of sheaves of complex-valued differential forms on X1 the underlying differentiable manifold is an acyclic resolution of C, and so Hn.X;C/D Hn.(cid:0).X;˝(cid:15) ˝C//; Hodge theory shows that each element of the second group is X1 representedbyauniqueharmonicn-form,andthedecompositioncorrespondstothedecom- positionofharmonicn-formsintosumsofharmonic.p;q/-forms,pCqDn. 3 Complete smooth varieties Finally,letX beacompletesmoothvarietyoverafieldk ofcharacteristiczero. IfkDC, thenX definesacompactcomplexmanifoldXan,andtherearethereforegroupsHn.Xan;Q/, dependingonthemapX !Spec.C/,thatweshallwriteHn.X/(hereB abbreviatesBetti). B IfX isprojective,thenthechoiceofaprojectiveembeddingdeterminesaKa¨hlerstructure on Xan, and hence a Hodge decomposition (which is independent of the choice of the embeddingbecauseitisdeterminedbytheHodgefiltration,andtheHodgefiltrationdepends onlyonX;seeTheorem1.4below). Inthegeneralcase,werefertoDeligne1968,5.3,5.5, fortheexistenceofthedecomposition. For an arbitrary field k and an embedding (cid:27)Wk ,!C, we write Hn.X/ for Hn.(cid:27)X/ (cid:27) B andHp;q.X/forHp;q.(cid:27)X/. As(cid:19)definesahomeomorphism(cid:27)Xan!(cid:19)(cid:27)Xan,itinducesan (cid:27) isomorphismHn.X/!Hn.X/. Sometimes,whenk isgivenasasubfieldofC,wewrite (cid:19)(cid:27) (cid:27) Hn.X/forHn.XC/. B B 3ForarecentaccountofHodgetheory,seeC.Voisin,HodgeTheoryandComplexAlgebraicGeometry,I, CUP,2002. 1 REVIEWOFCOHOMOLOGY 8 Let ˝(cid:15) denote the complex in which ˝n is the sheaf of algebraic differen- X=k X=k tial n-forms, and define the (algebraic) de Rham cohomology group Hn .X=k/ to be dR Hn.X ;˝(cid:15) / (hypercohomology with respect to the Zariski cohomology). For any Zar X=k homomorphism(cid:27)Wk,!k0,thereisacanonicalisomorphism Hn .X=k/˝ k0!Hn .X˝ k0=k0/: dR k;(cid:27) dR k Thespectralsequence Ep;q DHq.X ;˝p / H) HpCq.X ;˝(cid:15) / 1 Zar X=k Zar X=k definesafiltration(theHodgefiltration)FpHn .X/onHn .X/whichisstableunderbase dR dR change. THEOREM 1.4. WhenkDCtheobviousmaps Xan!X ; ˝(cid:15) ˝(cid:15); Zar Xan X induceisomorphisms Hn .X/!Hn .Xan/'Hn.Xan;C/ dR dR underwhichFpHn .X/correspondstoFpHn.Xan;C/Ddef M Hp0;q0. dR p0(cid:21)p,p0Cq0Dn PROOF. Theinitialtermsofthespectralsequences Ep;q DHq.X ;˝p / H) HpCq.X ;˝(cid:15) / 1 Zar X=k Zar X=k Ep;q DHq.X;˝p / H) HpCq.X;˝(cid:15) / 1 Xan Xan areisomorphic—seeSerre1956fortheprojectivecaseandGrothendieck1966forthegen- eralcase. Thetheoremfollowsfromthisbecause,bydefinitionoftheHodgedecomposition, thefiltrationofHn .Xan/definedbytheabovespectralsequenceisequaltothefiltrationof dR Hn.Xan;C/definedinthestatementofthetheorem. (cid:4) It follows from the theorem and the discussion preceding it that every embedding (cid:27)Wk,!Cdefinesanisomorphism ' HdnR.X/˝k;(cid:27)C(cid:0)!H(cid:27)n.X/˝QC and, in particular, a k-structure on H(cid:27)n.X/˝QC. When k DQ, this structure should be distinguishedfromtheQ-structuredefinedbyHn.X/: thetwoarerelatedbytheperiods. (cid:27) When k is algebraically closed, we write Hn.X;Af/, or Hent.X/, for Hn.Xet;ZO/˝Z Q, where Hn.X ;ZO/ D lim Hn.X ;Z=mZ/ (e´tale cohomology). If X is connected, et (cid:0) et m H0.X;A /DA , the ring of finite ade`les for Q, which justifies the first notation. By f f definition, Hn.X/ depends only on X (and not on its structure morphism X !Speck). et ThemapHn.X/!Hn.X˝ k0/definedbyaninclusionk,!k0 ofalgebraicallyclosed et et k fieldsisanisomorphism(specialcaseoftheproperbasechangetheoremArtinetal.1973, XII). The comparison theorem (ibid. XI) shows that, when k DC, there is a canonical isomorphismHn.X/˝A !Hn.X/. ItfollowsthatHn.X/˝A isindependentofthe B f et B f morphismX !SpecC,andthat,overanyalgebraicallyclosedfieldofcharacteristiczero, Hn.X/isafreeA -module. et f 1 REVIEWOFCOHOMOLOGY 9 TheA -moduleHn.X;A /canalsobedescribedastherestrictedproductofthespaces f f Hn.X;Q /,l aprimenumber,withrespecttothesubspacesHn.X;Z /=ftorsiong. l l Nextwedefinethenotionofthe“Tatetwist”ineachofthethreecohomologytheories. For this we shall define objects Q.1/ and set Hn.X/.m/DHn.X/˝Q.1/˝m. We want Q.1/tobeH2.P1/(realizationoftheTatemotiveinthecohomologytheory),buttoavoid thepossibilityofintroducingsignambiguitiesweshalldefineitdirectly, Q .1/D2(cid:25)iQ B Qet.1/DAf.1/Ddef .l im(cid:0)(cid:22)r/˝ZQ; (cid:22)r Df(cid:16) 2kj(cid:16)r D1g r Q .1/Dk; dR andso Hn.X/.m/DHn.X/˝Q.2(cid:25)i/mQDHn.Xan;.2(cid:25)i/mQ/ .kDC/ B B (cid:16) (cid:17) Hent.X/.m/DHent.X/˝Af .Af.1//˝mD l im(cid:0)rHn.Xet;(cid:22)˝r m/ ˝ZQ .k alg.closed) Hn .X/.m/DHn .X/. dR dR Thesedefinitionsextendinanobviouswaytonegativem. Forexample,wesetQ .(cid:0)1/D et HomAf.Af.1/;Af/anddefine Hn.X/.(cid:0)m/DHn.X/˝Q .(cid:0)1/˝m: et et et Therearecanonicalisomorphisms QB.1/˝QAf !Qet.1/ (k(cid:26)C,k algebraicallyclosed/ Q .1/˝C!Q .1/˝ C (k(cid:26)C) B dR k andhencecanonicalisomorphisms(thecomparisonisomorphisms) HBn.X/.m/˝QAf !Hent.X/.m/ (k(cid:26)C,k algebraicallyclosed/ HBn.X/.m/˝QC!HdnR.X/.m/˝kC (k(cid:26)C). Todefinethefirst,notethatexpdefinesanisomorphism z7!ezW2(cid:25)iZ=r2(cid:25)iZ!(cid:22) : r After passing to the inverse limit over r and tensoring with Q, we obtain the required isomorphism2(cid:25)iA !A .1/. Thesecondisomorphismisinducedbytheinclusions f f 2(cid:25)iQ,!C -k: AlthoughtheTatetwistfordeRhamcohomologyistrivial, itshouldnotbeignored. For example,whenkDC, Hn.X/˝C 17!.2(cid:25)i/m Hn.X/.m/˝C B ' B ' ' Hn .X/ Hn .X/.m/ dR dR 1 REVIEWOFCOHOMOLOGY 10 fails to commute by a factor .2(cid:25)i/m. Moreover when m is odd the top isomorphism is definedonlyuptosign. In each cohomology theory there is a canonical way of attaching a class cl.Z/ in H2p.X/.p/toanalgebraiccycleZ onX ofpurecodimensionp. Sinceourcohomology groupsarewithouttorsion,wecandothisusingChernclasses(Grothendieck1958). Starting withafunctorialisomorphismc WPic.X/!H2.X/.1/,oneusesthesplittingprincipleto 1 definetheChernpolynomial c .E/DPc .E/tp; c .E/2H2p.X/.p/; t p p ofavectorbundleE onX. ThemapE 7!c .E/isadditive,andthereforefactorsthrough t theGrothendieckgroupofthecategoryofvectorbundlesonX. But,asX issmooth,this groupisthesameastheGrothendieckgroupofthecategoryofcoherentO -modules,and X wecanthereforedefine 1 cl.Z/D c .O / .p(cid:0)1/Š p Z (loc. cit. 4.3). Indefiningc fortheBettiande´taletheories,webeginwithmaps 1 Pic.X/!H2.Xan;2(cid:25)iZ/ Pic.X/!H2.X ;(cid:22) / et r arisingasconnectinghomomorphismsfromthesequences 0!2(cid:25)i !O (cid:0)e(cid:0)x!p O(cid:2) !0 Xan Xan 0!(cid:22) !O(cid:2) !(cid:0)r O(cid:2) !0: r X X ForthedeRhamtheory,wenotethatthedlogmap,f 7! df ,definesamapofcomplexes f 0 O(cid:2) 0 (cid:1)(cid:1)(cid:1) X dlog O d ˝1 d ˝2 d (cid:1)(cid:1)(cid:1) X X X andhenceamap Pic.X/'H1.X;O(cid:2)/'H2.X;0!O(cid:2) !(cid:1)(cid:1)(cid:1)/ X X !H2.X;˝(cid:15)/DH2 .X/DH2 .X/.1/ X dR dR whose negative is c . It can be checked that the three maps c are compatible with the 1 1 comparisonisomorphisms(Deligne1971a,2.2.5.1),anditfollowsformallythatthemapscl arealsocompatibleonceonehascheckedthattheGysinmapsandmultiplicativestructures arecompatiblewiththecomparisonisomorphisms. When k D C, there is a direct way of defining a class cl.Z/ 2 H2d(cid:0)2pp.X.C/;Q/ (singular cohomology, d D dim.X/, p D codim.Z//: the choice of an i D (cid:0)1 deter- mines an orientation of X and of the smooth part of Z, and there is therefore a topo- logically defined class cl.Z/ 2 H2d(cid:0)2p.X.C/;Q/. This class has the property that for Œ!(cid:141)2H2d(cid:0)2p.X1;R/DH2d(cid:0)2p.(cid:0).X;˝(cid:15) //representedbytheclosedform!, X1 Z hcl.Z/;Œ!(cid:141)iD !. Z