Hodge Cycles on Abelian Varieties

65 Pages·2011·0.54 MB·English

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