Hodge Cycles On Abelian Varieties - James Milne
Hodge cycles on abelian varietiesP. Deligne (notes by J.S. Milne)July 4, 2003; June 1, 2011; October 1, 2018.AbstractThis is a TeXed copy of the article published inDeligne, P.; Milne, J.S.; Ogus, A.; Shih, Kuang-yen. Hodge cycles, motives,and Shimura varieties. LNM 900, Springer, 1982, pp. 9–100,M.1somewhat revised and updated. See the endnotes for more details. The originalarticle consisted of the notes (written by jsm) of the seminar “Périodes des IntégralsAbéliennes” given by P. Deligne at I.H.E.S., 1978-79. This file is available at n . . . . . . . . . . . . . . . . . . . . . . . .2Review of cohomology . . . . . . . . . . . . . . . . . . . .5Topological manifolds . .Differentiable manifolds .Complex manifolds . . .Complete smooth varietiesApplications to periods . .2.557711Absolute Hodge cycles; principle B . . . . . . . . . . . . . . .15Definitions (k algebraically closed of finite transcendence degree)Basic properties of absolute Hodge cycles . . . . . . . . .Definitions (arbitrary k) . . . . . . . . . . . . . . .Statement of the main theorem . . . . . . . . . . . . .Principle B . . . . . . . . . . . . . . . . . . . .35.Mumford-Tate groups; principle A . . . . . . . . . . . . . . .Characterizing subgroups by their fixed tensorsHodge structures . . . . . . . . . . .Mumford-Tate groups . . . . . . . . .Principle A . . . . . . . . . . . . .4.22.22232426Construction of some absolute Hodge cycles . . . . . . . . . . .28Hermitian forms . . . . . . . . . . . . . . . . . . . . . .VConditions for dE H 1 .A; Q/ to consist of absolute Hodge cycles . . . .2830Completion of the proof for abelian varieties of CM-type . . . . . .36Abelian varieties of CM-type. . . . . . . . . . . . . . . . . .Proof of the main theorem for abelian varieties of CM-type . . . . . . .36371.1516191919
CONTENTS672Completion of the proof; consequences . . . . . . . . . . . . .41Completion of the proof of Theorem 2.11 . . . . . . . . . . . . .Consequences of Theorem 2.11 . . . . . . . . . . . . . . . . .4142Algebraicity of values of the-function . . . . . . . . . . . . .The Fermat hypersurface . . . . . . . . .Calculation of the cohomology . . . . . . .The action of Gal.Q k/ on the étale cohomologyCalculation of the periods . . . . . . . . .The theorem . . . . . . . . . . . . . .Restatement of the theorem . . . . . . . .44.454649515355References. . . . . . . . . . . . . . . . . . . . . . . . . . .M Endnotes (by J.S. Milne) . . . . . . . . . . . . . . . . . . .5760IntroductionLet X be a smooth projective variety over C. Hodge conjectured that certain cohomologyclasses on X are algebraic. The main result proved in these notes shows that, when X is anabelian variety, the classes considered by Hodge have many of the properties of algebraicclasses.In more detail, let X an be the complex analytic manifold associated with X, and considerthe singular cohomology groups H n .X an ; Q/. The variety X an being of Kähler type (everyprojective embedding defines a Kähler structure), its cohomology groups H n .X an ; C/ 'H n .X an ; Q/ C have canonical decompositionsMdefpH n .X an ; C/ DH p;q ; H p;q D H q .X an ; X an /.pCqDnThe cohomology class cl.Z/ 2 H 2p .X an ; C/ of an algebraic subvariety Z of codimension pin X is rational (i.e., it lies in H 2p .X an ; Q// and is of bidegree .p; p/ (i.e., it lies in H p;p ).The Hodge conjecture states that, conversely, every element ofH 2p .X an ; Q/ \ H p;pis a Q-linear combination of the classes of algebraic subvarieties. Since the conjecture isunproven, it is convenient to call these rational .p; p/-classes Hodge cycles on X.Now consider a smooth projective variety X over a field k that is of characteristic zero,algebraically closed, and small enough to be embeddable in C. The algebraic de Rhamncohomology groups HdR.X k/ have the property that, for any embedding W k ,! C, thereare canonical isomorphismsnnHdR.X k/ k; C ' HdR.X an ; C/ ' H n .X an ; C/:2pIt is natural to say that t 2 HdR .X k/ is a Hodge cycle on X relative to if its image inH 2p .X an ; C/ is .2 i /p times a Hodge cycle on X k; C. The arguments in these notes2pshow that, if X is an abelian variety, then an element of HdR .X k/ that is a Hodge cycle onX relative to one embedding of k into C is a Hodge cycle relative to all embeddings; further,for any embedding, .2 i /p times a Hodge cycle in H 2p .X an ; C/ always lies in the image
CONTENTS32pof HdR .X k/.1 Thus the notion of a Hodge cycle on an abelian variety is intrinsic to thevariety: it is a purely algebraic notion. In the case that k D C the theorem shows that theimage of a Hodge cycle under an automorphism of C is again a Hodge cycle; equivalently,the notion of a Hodge cycle on an abelian variety over C does not depend on the mapX ! Spec C. Of course, all this would be obvious if only one knew the Hodge conjecture.In fact,Qa stronger result is proved in which a Hodge cycle is defined to be an element ofnHdR .X / l H n .Xet ; Ql /. As the title of the original seminar suggests, the stronger resulthas consequences for the algebraicity of the periods of abelian integrals: briefly, it allowsone to prove all arithmetic properties of abelian periods that would follow from knowing theHodge conjecture for abelian ��————In more detail, the main theorem proved in these notes is that every Hodge cycle on anabelian variety (in characteristic zero) is an absolute Hodge cycle — see 2 for the definitionsand Theorem 2.11 for a precise statement of the result.The proof is based on the following two principles.A. Let t1 ; : : : ; tN be absolute Hodge cycles on a smooth projective variety X and let G bethe largest algebraic subgroup of GL.H .X; Q// GL.Q.1// fixing the ti ; then everycohomology class t on X fixed by G is an absolute Hodge cycle (see 3.8).B. If .Xs /s2S is an algebraic family of smooth projective varieties with S connected andsmooth and .ts /s2S is a family of rational cycles (i.e., a global section of . . . ) suchthat ts is an absolute Hodge cycle for one s, then ts is an absolute Hodge cycle for alls (see 2.12, 2.15).Every abelian variety A with a Hodge cycle t is contained in a smooth algebraic familyin which t remains Hodge and which contains an abelian variety of CM-type. Therefore,Principle B shows that it suffices to prove the main theorem for A an abelian variety ofCM-type (see 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 withcomplexL multiplication by E. Principle B is used to construct some absolute Hodge classeson A2 A — the principle allows us to replace A by an abelian variety of the formA0 Z OE (see 4). Let G GL. A2 H1 .A; Q// GL.Q.1// be the subgroup fixing theabsolute Hodge cycles just constructed plus some other (obvious) absolute Hodge cycles.It is shown that G fixes every Hodge cycle on A, and Principle A therefore completes theproof (see 5).On analyzing which properties of absolute Hodge cycles are used in the above proof,one arrives at a slightly stronger result.2 Call a rational cohomology class c on a smoothprojective complex variety X accessible if it belongs to the smallest family of rationalcohomology classes such that:(a) the cohomology class of every algebraic cycle is accessible;(b) the pull-back by a map of varieties of an accessible class is accessible;(c) if t1 ; : : : ; tN 2 H .X; Q/ are accessible, and if a rational class t in some H 2p .X; Q/is fixed by an algebraic subgroup G of Aut.H .X; Q// (automorphisms of H .X; Q/as a graded algebra) fixing the ti , then t is accessible;1 Added(jsm).This doesn’t follow directly from Theorem 2.11 (see 2.4). However, one obtains a variant ofTheorem 2.11 using the above definitions simply by dropping the étale component everywhere in the proof (see,for example, 2.10b).2 Added(jsm). See also 9 of Milne, Shimura varieties and moduli, 2013.
CONTENTS4(d) Principle B holds with “absolute Hodge” replaced by “accessible”.Sections 4,5,6 of these notes can be interpreted as proving that, when X is an abelian variety,every Hodge cycle is accessible. Sections 2,3 define the notion of an absolute Hodge cycleand show that the family of absolute Hodge cycles satisfies (a), (b), (c), and (d); therefore,an accessible class is absolutely Hodge. We have the implications:M.3abelian varietiestrivialHodge HHHHHHHH) accessible HHHH) absolutely Hodge HHHH) Hodge.Only the first implication is restricted to abelian varieties.The remaining three sections, 1 and 7, serve respectively to review the differentcohomology theories and to give some applications of the main results to the algebraicity ofproducts of special values of the -function.N OTATIONWe define C to be the algebraic closure of Rpand i 2 C to be a square root of 1; thus iis only defined up to sign. A choice of i D1 determines an orientation of C as a realmanifold — we take that for which 1 i 0 — and hence an orientation of every complexmanifold. Complex conjugation on C is denoted by or by z 7! z.Recall that the category of abelian varieties up to isogeny is obtained from the categoryof abelian varieties by taking the same class of objects but replacing Hom.A; B/ withHom.A; B/ Q. We shall always regard an abelian variety as an object in the category ofabelian varieties up to isogeny: thus Hom.A; B/ is a vector space over Q.If .V / is a family of representations of an algebraic group G over a field k and t ;ˇ 2 V ,then the subgroup of G fixing the t ;ˇ is the algebraic subgroup H of G such that, for allk-algebras R,H.R/ D fg 2 G.R/ j g.t ;ˇ 1/ D t ;ˇ 1, all ; ˇg.Linear duals are denoted by . If X is a variety over a field k and is a homomorphism W k ,! k 0 , then X denotes the variety X k; k 0 (D X Spec.k/ Spec.k 0 /).By a b we mean that a is sufficiently greater than b.
11REVIEW OF COHOMOLOGY5Review of cohomologyTopological manifoldsLet X be a topological manifold and F a sheaf of abelian groups on X . We setH n .X; F / D H n . .X; F //where F ! F is any acyclic resolution of F . This defines H n .X; F / uniquely up to aunique isomorphism.When F is the constant sheaf defined by a field K, these groups can be identified withsingular cohomology groups as follows. Let S .X; K/ be the complex in which Sn .X; K/ isthe K-vector space with basis the singular n-simplices in X and the boundary map sends asimplex to the (usual) alternating sum of its faces. SetS .X; K/ D Hom.S .X; K/; K/with the boundary map for which. ; / 7! . /W S .X; K/ S .X; K/ ! Kis a morphism of complexes, namely, that defined by.d /. / D . 1/deg. /C1 .d /:P ROPOSITION 1.1. There is a canonical isomorphism H n .S .X; K// ! H n .X; K/.P ROOF. If U is the unit ball, then H 0 .S .U; K// D K and H n .S .U; K// D 0 for n 0.Thus, K ! S .U; K/ is a resolution of the group K. Let S n be the sheaf of X associatedwith the presheaf V 7! S n .V; K/. The last remark shows that K ! S is a resolution ofthe sheaf K. As each S n is fine (Warner 1971, 5.32), H n .X; K/ ' H n . .X; S //. But theobvious map S .X; K/ ! .X; S / is surjective with an exact complex as kernel (loc. cit.),and so'H n .S .X; K// ! H n . .X; S // ' H n .X; K/. Differentiable manifoldsNow assume X is a differentiable manifold. On replacing “singular n-simplex” by “differentiable singular n-simplex” in the above definitions, one obtains complexes S 1 .X; K/ and .X; K/. The same argument shows that there is a canonical isomorphismS1'nH1.X; K/ D H n .S 1 .X; K// ! H n .X; K/def(Warner 1971, 5.32).nLet OX 1 be the sheaf of C 1 real-valued functions on X, let X1 be the O X 1 -module 1of C differential n-forms on X, and let X 1 be the complexddd12O X 1 ! X1 ! X 1 ! :
1REVIEW OF COHOMOLOGY6The de Rham cohomology groups of X are defined to ben HdR.X / D H n . .X; X1 // Dfclosed n-formsg:fexact n-formsg0nIf U is the unit ball, Poincaré’s lemma shows that HdR.U / D R and HdR.U / D 0 for n 0. nThus, R ! X 1 is a resolution of the constant sheaf R, and as the sheaves X1 are finenn(Warner 1971, 5.28), we have H .X; R/ ' HdR .X /.n1For ! 2 .X; X1 / and 2 Sn .X; R/, defineZn.nC1/2h!; i D . 1/! 2 R. Stokes’s theorem states thatR d! DRd !, and sohd!; i C . 1/n h!; d i D 0.The pairing h; i therefore defines a map of complexes f W .X; X1 / ! S1 .X; R/.nn .X; R/ defined by f is an isomor.X / ! H1T HEOREM 1.2 ( DE R HAM ). The map HdRphism for all n.P ROOF. The map is inverse to the map'nnH1.X; R/ ! H n .X; R/ ' HdR.X /defined in the previous two paragraphs (Warner 1971, 5.36). (Our signs differ from the usualsigns because the standard sign conventionsZZZZZ d! D!;pr1 ! pr2 D! ; etc. d X YXviolate the sign conventions for complexes.)Y RA number !, 2 Hn .X; Q/, is called a period of !. The map in (1.2) identifiesH n .X; Q/ with the space of classes of closed forms whose periods are all rational. Theorem1.2 can be restated as follows: a closed differential form is exact if all its periods are zero;there exists a closed differential form having arbitrarily assigned periods on an independentset of cycles.R EMARK 1.3 (S INGER AND T HORPE 1967, 6.2). If X is compact, then it has a smoothtriangulation T . Define S .X; T; K/ and S .X; T; K/ as before, but using only simplices inT . Then the map .X; X1 / ! S .X; T; K/defined by the same formulas as f above induces isomorphismsnHdR.X / ! H n .S .X; T; K//.
1REVIEW OF COHOMOLOGY7Complex manifolds Now let X be a complex manifold, and let Xan denote the complexddd12OX an ! Xan ! X an ! nin which Xan is the sheaf of holomorphic differential n-forms. Thus, locally a section ofn X an is of the formX!D i1 :::in dzi1 : : : dzinwith i1 :::in a holomorphic function and the zi complex local coordinates. The complex form of Poincaré’s lemma shows that C ! Xan is a resolution of the constant sheaf C, andso there is a canonical isomorphism H n .X; C/ ! Hn .X; X(hypercohomology).an /If X is a compact Kähler manifold, then the spectral sequencep;qE1p D H q .X; X an / H) HpCq .X; Xan /degenerates, and so provides a canonical splittingMpH n .X; C/ DH q .X; X an / (the Hodge decomposition)pCqDndefpas H p;q D H q .X; X an / is the complex conjugate of H q;p relative to the real structureH n .X; R/ C ' H n .X; C/ (Weil 1958). The decomposition has the following explicit description: the complex X1 C of sheaves of complex-valued differential forms onthe underlying differentiable manifold is an acyclic resolution of C, and so H n .X; C/ D H n . .X; X1 C//; Hodge theory shows that each element of the second group isrepresented by a unique harmonic n-form, and the decomposition corresponds to the decomposition of harmonic n-forms into sums of harmonic .p; q/-forms, p C q D n. 3Complete smooth varietiesFinally, let X be a complete smooth variety over a field k of characteristic zero. If k D C,then X defines a compact complex manifold X an , and there are therefore groups H n .X an ; Q/,depending on the map X ! Spec.C/, that we shall write HBn .X / (here B abbreviates Betti).If X is projective, then the choice of a projective embedding determines a Kähler structureon X an , and hence a Hodge decomposition (which is independent of the choice of theembedding because it is determined by the Hodge filtration, and the Hodge filtration dependsonly on X ; see Theorem 1.4 below). In the general case, we refer to Deligne 1968, 5.3, 5.5,for the existence of the decomposition.For an arbitrary field k and an embedding W k ,! C, we write H n .X / for HBn . X /p;qand H .X / for H p;q . X /. As defines a homeomorphism X an ! X an , it induces ann .X / ! H n .X /. Sometimes, when k is given as a subfield of C, we writeisomorphism H nnHB .X / for HB .XC /.3 For a recent account of Hodge theory, see C. Voisin, Hodge Theory and Complex Algebraic Geometry, I,CUP, 2002.M.4
1REVIEW OF COHOMOLOGY8 nLet X kdenote the complex in which X kis the sheaf of algebraic differenntial n-forms, and define the (algebraic) de Rham cohomology group HdR.X k/ to be nH .XZar ; X k / (hypercohomology with respect to the Zariski cohomology). For anyhomomorphism W k ,! k 0 , there is a canonical isomorphismnnHdR.X k/ k; k 0 ! HdR.X k k 0 k 0 /:The spectral sequencep;qE1p D H q .XZar ; X k / H) HpCq .XZar ; X k/nndefines a filtration (the Hodge filtration) F p HdR.X / on HdR.X / which is stable under basechange.T HEOREM 1.4. When k D C the obvious mapsX an ! XZar ; Xan X;induce isomorphismsnnHdR.X / ! HdR.X an / ' H n .X an ; C/Mdefnunder which F p HdR.X / corresponds to F p H n .X an ; C/ D00H p ;q .p 0 p, p 0 Cq 0 DnP ROOF. The initial terms of the spectral sequencesp;qE1p D H q .XZar ; X k / H) HpCq .XZar ; X k/p;qE1p D H q .X; X an / H) HpCq .X; Xan /are isomorphic — see Serre 1956 for the projective case and Grothendieck 1966 for the general case. The theorem follows from this because, by definition of the Hodge decomposition,nthe filtration of HdR.X an / defined by the above spectral sequence is equal to the filtration ofnanH .X ; C/ defined in the statement of the theorem. It follows from the theorem and the discussion preceding it that every embedding W k ,! C defines an isomorphism'nHdR.X / k; C ! H n .X / Q Cand, in particular, a k-structure on H n .X / Q C. When k D Q, this structure should bedistinguished from the Q-structure defined by H n .X /: the two are related by the periods.O ZWhen k is algebraically closed, we write H n .X; Af /, or Hetn .X /, for H n .Xet ; Z/nnOQ, where H .Xet ; Z/ D lim H .Xet ; Z mZ/ (étale cohomology). If X is connected,mH 0 .X; Af / D Af , the ring of finite adèles for Q, which justifies the first notation. Bydefinition, Hetn .X / depends only on X (and not on its structure morphism X ! Spec k).The map Hetn .X / ! Hetn .X k k 0 / defined by an inclusion k ,! k 0 of algebraically closedfields is an isomorphism (special case of the proper base change theorem Artin et al. 1973,XII). The comparison theorem (ibid. XI) shows that, when k D C, there is a canonicalisomorphism HBn .X / Af ! Hetn .X /. It follows that HBn .X / Af is independent of themorphism X ! Spec C, and that, over any algebraically closed field of characteristic zero,Hetn .X / is a free Af -module.
1REVIEW OF COHOMOLOGY9The Af -module H n .X; Af / can also be described as the restricted product of the spacesH n .X; Ql /, l a prime number, with respect to the subspaces H n .X; Zl / ftorsiong.Next we define the notion of the “Tate twist” in each of the three cohomology theories.For this we shall define objects Q.1/ and set H n .X /.m/ D H n .X / Q.1/ m . We wantQ.1/ to be H 2 .P1 / (realization of the Tate motive in the cohomology theory), but to avoidthe possibility of introducing sign ambiguities we shall define it directly,QB .1/ D 2 i Q r D f 2 k j r D 1gdefQet .1/ D Af .1/ D .lim r / Z Q;rQdR .1/ D k;and soHBn .X /.m/ D HBn .X / Q .2 i /m Q D H n .X an ; .2 i /m Q/ .k D C/ Hetn .X /.m/ D Hetn .X / Af .Af .1// m D lim r H n .Xet ; m/ Z Q .k alg. closed)rnnHdR.X /.m/ D HdR.X /.These definitions extend in an obvious way to negative m. For example, we set Qet . 1/ DHomAf .Af .1/; Af / and defineHetn .X /. m/ D Hetn .X / Qet . 1/ m :There are canonical isomorphismsQB .1/ Q Af ! Qet .1/ (k C, k algebraically closed/QB .1/ C ! QdR .1/ k C(k C)and hence canonical isomorphisms (the comparison isomorphisms)HBn .X /.m/ Q Af ! Hetn .X /.m/ (k C, k algebraically closed/nHBn .X /.m/ Q C ! HdR.X /.m/ k C(k C).To define the first, note that exp defines an isomorphismz 7! e z W 2 i Z r2 iZ ! r :After passing to the inverse limit over r and tensoring with Q, we obtain the requiredisomorphism 2 i Af ! Af .1/. The second isomorphism is induced by the inclusions2 iQ ,! C- k:Although the Tate twist for de Rham cohomology is trivial, it should not be ignored. Forexample, when k D C,HBn .X / C'nHdR.X /17!.2 i /m'HBn .X/.m/ C'nHdR.X /.m/
1REVIEW OF COHOMOLOGY10fails to commute by a factor .2 i /m . Moreover when m is odd the top isomorphism isdefined only up to sign.In each cohomology theory there is a canonical way of attaching a class cl.Z/ in2pH .X /.p/ to an algebraic cycle Z on X of pure codimension p. Since our cohomologygroups are without torsion, we can do this using Chern classes (Grothendieck 1958). Startingwith a functorial isomorphism c1 W Pic.X / ! H 2 .X /.1/, one uses the splitting principle todefine the Chern polynomialPct .E/ D cp .E/t p ; cp .E/ 2 H 2p .X /.p/;of a vector bundle E on X . The map E 7! ct .E/ is additive, and therefore factors throughthe Grothendieck group of the category of vector bundles on X . But, as X is smooth, thisgroup is the same as the Grothendieck group of the category of coherent OX -modules, andwe can therefore define1cl.Z/ Dcp .OZ /.p 1/Š(loc. cit. 4.3).In defining c1 for the Betti and étale theories, we begin with mapsPic.X / ! H 2 .X an ; 2 i Z/Pic.X / ! H 2 .Xet ; r /arising as connecting homomorphisms from the sequencesexp0 ! 2 i ! OX an ! O X an ! 0r 0 ! r ! O X ! O X ! 0:For the de Rham theory, we note that the d log map, f 7! OX0dff, defines a map of complexes 0dlogOXd1 Xd2 Xd and hence a map2 Pic.X / ' H 1 .X; O X / ' H .X; 0 ! O X ! / 22! H2 .X; X/ D HdR.X / D HdR.X /.1/whose negative is c1 . It can be checked that the three maps c1 are compatible with thecomparison isomorphisms (Deligne 1971a, 2.2.5.1), and it follows formally that the maps clare also compatible once one has checked that the Gysin maps and multiplicative structuresare compatible with the comparison isomorphisms.When k D C, there is a direct way of defining a class cl.Z/ 2 H2d 2p .X.C/; Q/p(singular cohomology, d D dim.X /, p D codim.Z//: the choice of an i D1 determines an orientation of X and of the smooth part of Z, and there is therefore a topologically defined class cl.Z/ 2 H2d 2p .X.C/; Q/. This class has the property that for Œ! 2 H 2d 2p .X 1 ; R/ D H 2d 2p . .X; X1 // represented by the closed form !,Zhcl.Z/; Œ! i D!.Z
1REVIEW OF COHOMOLOGY112pBy Poincaré duality, cl.Z/ corresponds to a class cltop .Z/ 2 HB .X /, whose image in2pHB .X /.p/ under the map induced by 1 7! .2 i /p W Q ! Q.p/ is known to be clB .Z/. Theabove formula becomesZZcltop .Z/ [ Œw D!.XZThere are trace maps (d D dim X )'TrB W HB2d .X /.d / ! Q'Tret W Het2d .X /.d / ! Af'2dTrdR W HdR.X /.d / ! kthat are determined by the requirement that Tr.cl.point// D 1. They are compatible withthe comparison isomorphisms. When k D C, TrB and TrdR are equal respectively to thecompositesHB2d .X/.d /.2 i /d 7!12dHdR.X/.d /Œ! 7!RHB2d .X /'H 2d . . X1 //2dHdR.X /'H 2d .Œ! 7! 1 d.2 i/ . X1 //X!CRX!Cwhere we have chosen an i and used it to orientate X (the composite maps are obviouslyindependent of the choice of i ). The formulas of the last paragraph show thatZ1TrdR .cldR .Z/ [ Œ! / D!:.2 i /dim Z ZA definition of Tret can be found in Milne 1980, VI 11.Applications to periodsWe now deduce some consequences concerning periods.P ROPOSITION 1.5. Let X be a complete smooth variety over an algebraically closed fieldk C and let Z be an algebraic cycle on XC of dimension r. For any C 1 differential2r2rr-form ! on XC whose class Œ! in HdR.XC / lies in HdR.X /Z! 2 .2 i /r k:ZP ROOF. We first note that Z is algebraically equivalent to a cycle Z0 defined over k.In proving this, we can assume Z to be prime. There exists a smooth variety T over k, asubvariety Z X T that is flat over T , and a point Spec C ! T such that Z D Z T Spec Cin X T Spec C D XC . We can therefore take Z0 to be Z T Spec k X T Spec k D X2rfor any point Spec k ! T . From this it follows thatdR .Z0 / 2 HdR .X /.r/ andR cldR .Z/ D clrTrdR .cldR .Z/ [ Œ! / 2 k. But we saw above that Z ! D .2 i / TrdR .cldR .Z/ [ Œ! /. We next derive a classical relation between the periods of an elliptic curve. For acomplete smooth curve X and an open affine subset U , the map11HdR.X / ! HdR.U / D1.U; X/fmeromorphic diffls, holomorphic on U gDd .U; OX /fexact differentials on U g
1REVIEW OF COHOMOLOGY12is injective with image the set of classes represented by forms whose residues are all zero(such forms are said to be of the second kind). When k D C, TrdR .Œ [ Œˇ /, where and ˇare differential 1-forms of the second kind, can be computed as follows. Let be the finiteset of points where or ˇ has a pole. For z a local parameter at P 2 , can be writtenX Dai z i dz with a 1 D 0:1 i 1ThereRtherefore exists a meromorphic function f defined near P such that df D .R Wewrite for any such function — it is defined up to a constant. As ResP ˇ D 0, ResP . /ˇis well-defined, and one proves thatXR TrdR .Œ [ Œˇ / DResP ˇ.P 2 Now let X be the elliptic curvey 2 z D 4x 3g2 xz 2g3 z 3 :There is a lattice in C and corresponding Weierstrass function }.z/ such thatz 7! .}.z/ W } 0 .z/ W 1/defines an isomorphism C ! X.C/. Let 1 and 2 be generators of such that the basesf 1 ; 2 g and f1; i g of C have the same orientation. We can regard 1 and 2 as elements ofH1 .X; Z/, and then 1 2 D 1. The differentials ! D dx y and D xdx y on X pull backto dz and }.z/dz respectively on C. The first is therefore holomorphic and the second has asingle pole at 1 D .0 W 1 W 0/ on X with residue zero (because 0 2 C maps to 1 2 X and}.z/ D z12 C a2 z 2 C : : :). We find that Z TrdR .Œ! [ Œ / D Res0dz }.z/ dz D Res0 .z}.z/ dz/ D 1.For i D 1; 2, letdx defDyZx dx defDyZZiZidxp4x 3g2 xxdxg3p4x 3g2 xg3iiD !iD ibe the periods of ! and . Under the map1HdR.X / ! H 1 .X; C/! maps to !1f 1 ; 2 g. Thus001 C !2 2and maps to 1001 C 2 2 ,where f 10 ;1 D TrdR .Œ! [ Œ /D TrB .!100001 C !2 2 / [ . 1 1 C 2 2 //!2 1 / TrB . 10 [ 20 /D .!1 21D.!1 22 i!2 1 /:02gis the basis dual to
1REVIEW OF COHOMOLOGY13Hence! 1 2!2 1 D 2 i.This is the Legendre relation.The next proposition shows how the existence of algebraic cycles can force algebraicrelations between the periods of abelian integrals. Let X be an abelian variety over a subfieldk of C. In each of the three cohomology theories, rH r .X / DH 1 .X /andH 1 .X X / D H 1 .X / H 1 .X / Let 2 Gm .Q/ act on QB .1/ as 1 . There is then a natural action of GL.HB1 .X // Gm onHBr .X n /.m/ for any r; n; and m. We define G to be the subgroup of GL.HB1 .X // Gm fixingall the tensors of the form clB .Z/, Z an algebraic cycle on some X n (see the Notations).Consider the comparison isomorphisms'1HdR.X / k C ! H 1 .X an ; C/'HB1 .X / Q C.The periods pij of X are defined by the equationsX i Dpj i aj1where f i g and fai g are bases for HdR.X / and HB1 .X / over k and Q respectively. The fieldk.pij / generated over k by the pij is independent of the bases chosen.P ROPOSITION 1.6. With the above definitions, the transcendence degree of k.pij / over kis dim.G/.P ROOF. We can replace k by its algebraic closure in C, and hence assume that each algebraiccycle on XC is equivalent to an algebraic cycle on X (see the proof of 1.5). Let P be the1functor of k-algebras whose value on R is the set of isomorphisms pW HB1 Q R ! HdR k Rmapping clB .Z/ 1 to cldR .Z/ 1 for all algebraic cycles Z on a power of X . WhenR D C, the comparison isomorphism is such a p, and so P .C/ is not empty. It is easily seenthat P is represented by an algebraic variety that becomes a Gk -torsor under the obviousaction. The bases f i g and fai g can be used to identify the points of P with matrices. Thematrix pij is a point of P with coordinates in C, and so the proposition is a consequenceof the following easy lemma. L EMMA 1.7. Let AN be the affine N -space over a subfield k of C, and let z 2 AN .C/. Thetranscendence degree of k.z1 ; : : : ; zN / over k is the dimension of the Zariski closure of fzgin AN .P ROOF. Let a be the kernel of the homomorphism kŒT1 ; : : : ; TN ! kŒz1 ; : : : ; zN sendingTi to zi . The Zariski closure of fzg in AN is the zero set Z.a/ of a, and Z.a/ is a varietyover k with function field k.z1 ; : : : ; zN /. Now dim.Z.a// D tr. deg.k k.z1 ; : : : ; zN / (standardresult).
1REVIEW OF COHOMOLOGY14R EMARK 1.8. If X is an elliptic curve, then dim G is 2 or 4 according as X has complexmultiplication or not. Chudnovsky has shown thattr. deg.k k.pij / D dim Gwhen X is an elliptic curve with complex multiplication. Does equality hold for all abelianvarieties?M.5One of the main purposes of the seminar was to show that, in the case that X is anabelian variety, (1.5) and (1.6) make sense, and remain true, if “algebraic cycle” is replacedby “Hodge cycle”.
22ABSOLUTE HODGE CYCLES; PRINCIPLE B15Absolute Hodge cycles; principle BDefinitions (k algebraically closed of finite transcendence degree)Let k be an algebraically closed field of finite transcendence degree over Q, and let X be acomplete smooth variety over k. SetnHAn .X /.m/ D HdR.X /.m/ Hetn .X /.m/— it is a free k Af -module. Corresponding to an embedding W k ,! C, there are canonicalisomorphisms' nn dRWHdR.X /.m/ k; C ! HdR. X /.m/' et WHetn .X /.m/ ! Hetn . X /.m/whose product we write . The diagonal embeddingnH n .X /.m/ ,! HdR. X /.m/ Hetn . X /.m/induces an isomorphism'n. X /.m/ Hetn . X /.m/H n .X /.m/ .C Af / ! HdR2p(product of the comparison isomorphisms, 1). An element t 2 HA .X /.p/ is a Hodgecycle relative to if2p(a) t is rational relative to , i.e., .t / lies in the rational subspace H .X /.p/ of2p2pHdR . X /.p/ Het . X /.p/;2p2pdef(b) the first component of t lies in F 0 HdR .X /.p/ D F p HdR .X /.2pEquivalent condition: .t / lies in H .X /.p/ and is of bidegree .0; 0/. If t is a Hodgecycle relative to every embedding W k ,! C, then i
Hodge cycles on abelian varieties P. Deligne (notes by J.S. Milne) July 4, 2003; June 1, 2011; October 1, 2018. Abstract This is a TeXed copy of the article published in
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Quantization of Gauge Fields We will now turn to the problem of the quantization of gauge th eories. We will begin with the simplest gauge theory, the free electromagnetic field. This is an abelian gauge theory. After that we will discuss at length the quantization of non-abelian gauge fields. Unlike abelian theories, such as the
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Sonia L’Innocente Abelian Regularization of Rings and Modules. Our context Main Results Relation between Cohn and Ziegler Spectrum If R is abelian regular, then the points of the Ziegler spectrum are given by the endosimple mod
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