ON THE IMAGE OF THE l-ADIC ABEL-JACOBI MAP FOR A VARIETY OVER THE ALGEBRAIC ...

44 Pages·1999·0.66 MB·English

JOURNALOFTHE
AMERICANMATHEMATICALSOCIETY
Volume12,Number3,Pages795{838
S0894-0347(99)00303-3
ArticleelectronicallypublishedonApril23,1999
ON THE IMAGE OF THE l-ADIC ABEL-JACOBI MAP
FOR A VARIETY OVER THE ALGEBRAIC CLOSURE
OF A FINITE FIELD
CHAD SCHOEN
0. Introduction
Let k0 be a (cid:12)nite (cid:12)eld of characteristic p whose algebraic closure we denote by
k(cid:22). Let Y be a smooth projective k0-variety. For each prime l6=p there is an l-adic
Abel-Jacobi map
arY;l :CHhrom(Yk(cid:22))!H2r−1(Yk(cid:22);Zl(r))⊗Ql=Zl;
which is a potentially useful tool for studying the Chow group of nullhomologous
cycles on Y(cid:22). In this paper we investigate when ar is surjective. The (cid:12)rst two
k Y;l
results are conditional, because they depend on assuming the truth of the Tate
conjecture (0.7).
(0.1) Theorem. Suppose that (0.7) holds. If p > 2 and the dimension of Y is at
most 4, then the set of prime numbers
L :=fl prime:l 6=p and ar is not surjective for some rg
Y Y;l
is finite.
In order to extend (0.1) to varieties of arbitrary dimension we (cid:12)nd it necessary
to make an assumption which goes beyond the Tate conjecture. Since such an
assumption is easily formulated, we include it in x9.6 as Hypothesis H. With this
hypothesis the same arguments which prove(0.1) yieldimmediately a result which
is independent of the dimension of Y:
(0.2) Theorem. Suppose that (0.7) and Hypothesis H hold. If p > 2, then L is
Y
a finite set.
To obtain results which don’t depend on conjectures, we restrict attention to
varieties of a special form. Let (cid:25) :Y !X be a non-isotrivial, semi-stable, elliptic
surfacewithasectionde(cid:12)nedovera(cid:12)nite(cid:12)eldk0 ofcharacteristicp>2. Writem(cid:25)
for the least common multiple of all m such that (cid:25) has a singular (cid:12)ber of Kodaira
type I . Let W be the non-singularvariety obtainedby blowingup Y (cid:2) Y along
m X
the singular locus.
(0.3) Theorem. If l (cid:45)2(cid:1)5(cid:1)p(cid:1)m , then a2 is surjective.
(cid:25) W;l
ReceivedbytheeditorsJune24,1997and,inrevisedform,January5,1999.
1991 Mathematics Subject Classification. Primary14C25,14G15.
Key words and phrases. Algebraiccycles,l-adicAbel-Jacobimap.
ThisresearchwaspartiallysupportedbyNSFgrantsDMS-90-14954,DMS-93-06733.
(cid:13)c1999 American Mathematical Society
795
796 CHAD SCHOEN
An application of (0.3) and a theorem of Soul(cid:19)e [So] is the following extension of
[Sch, 14.2]:
(0.4) Theorem. Let E (cid:26) P2 be the Fermat cubic curve. If l (cid:45) 2(cid:1)3(cid:1)p, then a2
E3;l
gives rise to an isomorphism
CHh2om(EF(cid:22)3p)⊗Zl !H3(EF(cid:22)3p;Ql=Zl(2)):
This last result may be reformulated in terms of the coniveau (cid:12)ltration on the
third cohomology. The interesting piece of this (cid:12)ltration is
NH3(Yk(cid:22);Ql=Zl(2)):=Ker [H3(Yk(cid:22);Ql=Zl(2))!H3(k(cid:22)(Y);Ql=Zl(2))]:
From work of Bloch and Merkuriev and Suslin one knows that
CHh2om(Yk(cid:22))tors⊗Zl ’NH3(Yk(cid:22);Ql=Zl(2));
when H4(Yk(cid:22);Zl(2)) is torsion free (cf. [Ras, 3.6]). Thus (0.4) is equivalent to
(0.5) NH3(EF(cid:22)3p;Ql=Zl(2))=H3(EF(cid:22)3p;Ql=Zl(2))
for l(cid:45)2(cid:1)3(cid:1)p.
Itisinterestingtospeculateif(0.5)stillholdswhenE3isreplacedbyanysmooth,
projectiveY andlisallowedtobeanyprimedi(cid:11)erentfromp. Anecessarycondition
for this to happen would be the surjectivity of a2 8 l 6= p. However, this would
Y;l
not be su(cid:14)cient. In addition one would have to show that the torsion subgroup of
CH2 (Y(cid:22)) is large. Soul(cid:19)e showed that CH2 (Y(cid:22)) is a torsion group when Y(cid:22) is
hom k hom k k
a three-dimensional Abelian variety [So, 3.3]. One might hope that this result will
eventually be extended to all smooth, projective varieties Y(cid:22), although substantial
k
progress in this direction is not known to the author. The group CHh2om(Yk(cid:22))tors,
for certain special varieties Y(cid:22), is the subject of [Sch-T].
k
All the main results of this paper depend strongly on the assumption that the
base (cid:12)eld, k(cid:22), is the algebraic closure of a (cid:12)nite (cid:12)eld. Analogous assertions about
smooth, projective varieties over other algebraically closed (cid:12)elds are often false.
For example, Bloch and Esnault considered the case of a variety, Y, de(cid:12)ned over
a number (cid:12)eld with good, ordinary reduction at a place above l and satisfying
H0(Y;Ω3)6=0. They proved that
Y
(0.6) NH3(YQ(cid:22);Ql=Zl(2))6=H3(YQ(cid:22);Ql=Zl(2))
whenever certain technical hypotheses hold [Bl-Es]. In particular, (0.6) holds if
l (cid:17) 1 mod 3 and YQ(cid:22) = EQ3(cid:22), where E is the Fermat cubic curve. This provides an
interesting contrast with (0.5).
Furthercontrastsbecomeevidentwhenonecomparestheresultsdescribedabove
with theorems and conjectures concerning the Hodge theoretic Abel-Jacobi map,
arY : CHhrom(YC) ! Jr(YC), for varieties de(cid:12)ned over C. This map is known not
to be surjective when Fr+1H2r−1(Y(C);C) 6= 0 [Gri, 13.2]. In fact, Green [Gre]
and Voisin (unpublished) show that the image of a2 is torsion for a su(cid:14)ciently
V
general hypersurface, VC (cid:26) P4C, of high degree. Combining this with the work of
Bloch and Esnault [Bl-Es, 4.1] leads one to speculate that a2 might be the zero
V
map. A conjecture of Nori may be viewed as going even further to suggest that
CHh2om(VC)=0 [No, 7.2.5].
Before outlining the organization of the paper we state the Tate conjecture in
the form required for (0.1) and (0.2):
THE IMAGE OF THE l-ADIC ABEL-JACOBI MAP 797
(0.7) Conjecture. Let k0 (cid:26)k be an arbitrary (cid:12)nite extension and let V=k be an
arbitrary smooth, projective variety. Then L
(0.7.1) the Frobenius element (cid:30)2Gk acts semi-simply on j(cid:21)0Hj(Vk(cid:22);Ql) and
(0.7.2) the cycle class map CHs(V)⊗Ql ! H2s(Vk(cid:22);Ql(s))Gk is surjective for
all s.
Wenowdescribetheorganizationofthepaperandthecontentsoftheindividual
sections. The(cid:12)rstsectionisdevotedtode(cid:12)nitionsandbasicpropertiesofthel-adic
intermediate Jacobian and the l-adic Abel-Jacobi map, ar . We also consider a
Y;l
modl versionofar whichiseasiertocompute,butmaystillbeusedtoshowthat
Y;l
ar is surjective.
Y;l
Theproofsof(0.1)and(0.2)makeuseofthetheoryofLefschetzpencils. Central
to the argument is the study of cycles supported in the (cid:12)bers of such a pencil.
Thus in x2 we consider a modi(cid:12)ed l-adic Abel-Jacobi map which is suitable for
studying cycles supported in the (cid:12)bers of a morphism from a variety to a curve.
The point here is to reduce the problem of evaluating ar at a nullhomologous
Y;l
cycle to considerations involving only constructible sheaves on the curve. In fact
this variantof the l-adic Abel-Jacobi map may be evaluated on Tate classes in the
cohomology of the (cid:12)bers, even if these are not known to correspond to algebraic
cycles. We de(cid:12)ne a mod l version of this map and observe that evaluating it on a
Tateclasscorrespondstocomputingacoboundarymapassociatedwiththerelative
cohomology sequence for a certain constructible sheaf, M, of F -vector spaces on
l
the curve.
Sections 3 through 7 are devoted to the study of this coboundary map. The
challenge here is to identify a (cid:12)nite (cid:12)eld extension, k0 (cid:26)k, and a k rational point,
x2X , such that the coboundary map
k
Hx2(Xk(cid:22);M)Gk !H1(Gk;H1(Xk(cid:22);M))
isinjective. Furthermorethedomainofthismapshouldcontainanon-zeroelement
which comes from a Tate class. Requiring that H2(X(cid:22);M)Gk be one-dimensional
x k
turns out to be helpful here. The main technical result is Theorem (3.4).
The proof of Theorem (3.4) is begun in x4. We (cid:12)rst pass to a (cid:12)nite Galois
cover of the base curve in order to trivialize the sheaf M. Write Γ for the Galois
group of this cover. The original coboundary map may now be replaced with a
F [Γ]-linear coboundary map involving the cohomology of a constant sheaf on the
l
covering curve.
Sections 5 and 6 are devoted to the detailed study of the cohomology of the
coveringcurve as a module over F [Γ]. The proof of Theorem (3.4) is completed in
l
x7.
The next section contains a criterion for the existence of many Tate classes in
the cohomology of the (cid:12)bers of a morphism from a variety to a curve. In (8.2.2)
the connection between H2(X(cid:22);M)Gk being one-dimensional and Tate classes is
x k
discussed.
Section9isdevotedtoshowingthatthemachinerydevelopedinprevioussections
to evaluate the mod l Abel-Jacobi map at Tate classes in (cid:12)bers applies to certain
Lefschetzpencils. InparticularwecheckthatthereareplentyofTateclassesinthe
(cid:12)bersandthatthehypothesesofTheorem(3.4)arefull(cid:12)lledwhenMisconstructed
from the vanishing cohomology.
798 CHAD SCHOEN
The proofs of (0.1) and (0.2) are given in x10. We start with the case of curves,
where the results are well known, and proceed by induction on the dimension. A
givenvarietyY willbeblownupalongthebaselocusofaLefschetzpenciltoobtain
a variety W which maps to a curve. If the dimension of Y is odd, the techniques
developed in the (cid:12)rst part of the paper enable us to study the l-adic Abel-Jacobi
mapforthemiddledimensionalcohomology. ByinductionandtheTateconjecture
this is the crucial case.
Theorem (0.3) is a generalization of [Sch, 0.4]. It is proved in x11 by recalling
facts about elliptic surfaces and complex multiplication cycles and then applying
Theorem(3.4). Now Theorem (0.4) follows from (0.3) much as [Sch, 14.2] followed
from [Sch, 10.2]. The reader interested only in the proofs of (0.3) and (0.4) may
skip x8, x9 and x10.
I wish to thank M.S.R.I., the Max-Planck-Institutfu¨r Mathematik in Bonn and
the I.H.E.S. for their hospitality while various parts of this work were being done.
I also thank the referee for his or her comments.
Notationalconventions. Fora(cid:12)eldK,K(cid:22) denotesaseparableclosureandG :=
K
Gal(K(cid:22)=K).
Variety means a geometrically integral, separated scheme of (cid:12)nite type over a
(cid:12)eld. A curve is a variety of dimension 1.
k0 (cid:26)k1 (cid:26)k are (cid:12)nite (cid:12)elds.
l is an odd prime number, which is distinct from the characteristic of the base
(cid:12)eld. F is a (cid:12)eld with l elements.
l
If H isanAbelian group,H=t denotes the quotientofH by its torsionsubgroup
and H[n] denotes the kernel of multiplication by n.
Zr(W) denotes the group of codimension r algebraic cycles on a variety W.
If W is smooth over a (cid:12)eld K of characteristic p(cid:21)0, de(cid:12)ne
Y
Zhrom(W):=Ker[Zr(W)! H2r(WK(cid:22);Zl(r))]:
l6=p
For V a closed subscheme of a variety W and F on W,
HVi (W;F)0 :=Ker[HVi (W;F)!Hi(W;F)]:
If (cid:3) is a short exact sequence of G -modules, then the (cid:12)rst coboundary map in
k
the associated long exact Gk-cohomology sequence will be denoted (cid:14)(cid:3).
IfM isagrouprepresentation,thecontragredientrepresentationwillbedenoted
M_.
1. The l-adic intermediate Jacobian and the l-adic Abel-Jacobi map
1.1. The l-adic intermediate Jacobian. Let W be a variety which is proper
and smooth over a (cid:12)eld K0 which is (cid:12)nitely generated over the prime (cid:12)eld. Fix a
separable closure K(cid:22) of K0 and let l be a prime distinct from the characteristic of
K0. For an Abelian group H, H=t denotes the quotient by the torsion subgroup.
We de(cid:12)ne the l-adic intermediate Jacobian
(1.1.1) Jlr(W):=−li!mK H1(GK;H2r−1(WK(cid:22);Zl(r))=t);
where the direct limit is taken over sub(cid:12)elds K (cid:26)K(cid:22) which are (cid:12)nite extensions of
K0. InthispaperweshallbeexclusivelyconcernedwiththecaseinwhichK0 =k0
is a (cid:12)nite (cid:12)eld. In this situation the structure of Jr(W) turns out to be especially
l
simple.
THE IMAGE OF THE l-ADIC ABEL-JACOBI MAP 799
It is useful to introduce the more general notion of the l-adic intermediate Ja-
cobian of a Galois module. Let H be a (cid:12)nitely generated Z -module on which the
l
absolute Galois group G acts continuously. Suppose further that no eigenvalue
k0
for the operation of the Frobenius element (cid:30) 2 G on H ⊗Q is a root of unity.
k0 l
SuchG -modulesformacategoryandwemayconsiderthefunctortothecategory
k0
of Abelian groups de(cid:12)ned by
1
(1.1.2) J(H):=l−i!mkH (Gk;H=t);
where the limit is over intermediate (cid:12)elds k0 (cid:26)k (cid:26)k(cid:22) of (cid:12)nite degree over k0.
(1.1.3) Lemma. (i) J(H2r−1(Wk(cid:22);Zl(r))) is defined and is isomorphic to Jlr(W).
(ii) J(H)’H⊗Q =Z as G -modules.
l l k0
(iii) For each finite extension k0 (cid:26)k, J(H)Gk is a finite group.
Proof. (i) By Deligne’s theorem [De1] the eigenvalues of the arithmetic Frobenius,
(cid:30), acting on H2r−1(Wk(cid:22);Zl(r))⊗Ql are algebraic numbers with complex absolute
value di(cid:11)erentfrom1. The (cid:12)rststep in the proofof(ii) is to applyG -cohomology
k
to the short exact sequence
(1.1.4) 0!H=t !H⊗Ql !H⊗Ql=Zl !0:
A topological generator of G is a power, (cid:30)m, of (cid:30). Since no root of unity is an
k
eigenvalue of (cid:30), both
H0(Gk;H⊗Ql)=Ker(IdH⊗Ql −(cid:30)m)
and
H1(Gk;H⊗Ql)=Coker(IdH⊗Ql −(cid:30)m)
are zero. Now assertion (ii) follows by taking direct limits. For (iii) observe that
(H ⊗Q =Z )Gk is (cid:12)nite, since 1 is not an eigenvalue for the action of (cid:30)m on H ⊗
l l
Q .
l
For future reference we record four additional facts related to the functor J.
(1.1.5) Lemma. (i) Let h : H ! H0 be a homomorphism of finitely generated
Z-modules. Then h⊗Id:H⊗Q !H0⊗Q is surjective iff h⊗Id:H⊗Q =Z !
l l l l l
H0⊗Q =Z is.
l l
(ii) For any finite extension k=k0, the natural map H1(Gk;H=t) ! J(H)Gk is
an isomorphism.
(iii) If H is torsion free, then there is a canonical isomorphism, J(H)Gk=l !
H1(G ;H=l).
k
(iv) Let h : H ! H0 be a homomorphism of finitely generated Z -modules and
l
let A (cid:26) H ⊗Q =Z be a subgroup. If the Tate module of A tensored with Q maps
l l l
surjectively to H0⊗Q , then A maps surjectively to H0⊗Q =Z.
l l l
Proof. (i) follows from (1.1.4). (ii) follows from (1.1.3)(ii) and (1.1.4). For (iii)
apply G -cohomology to
k
0!H⊗ 1Z =Z !H⊗Q =Z −−−l−! H⊗Q =Z !0:
l l l l l l l
For(iv)notethatAisthedirectsumofitsmaximaldivisiblesubgroupanda(cid:12)nite
group. The former may be written H00⊗Q =Z where H00 is a Z -submodule of H.
l l l
Now the tautological map H00 ⊗Q =Z ! H0 ⊗Q =Z is surjective if and only if
l l l l
H00⊗Q !H0⊗Q is surjective.
l l
800 CHAD SCHOEN
1.2. The de(cid:12)nition of the l-adic Abel-Jacobi map. For each(cid:12)nite extension
k=k0 there is an l-adic Abel-Jacobi map
(1.2.1) (cid:11)r :Zr (W )!Jr(W)Gk:
l;k hom k l
To evaluate (cid:11)r on a cycle z write
l;k
(1.2.2) bzc2Hj2zrj(Wk(cid:22);Zl(r))0 :=Ker[Hj2zrj(Wk(cid:22);Zl(r))!H2r(Wk(cid:22);Zl(r))]
forthe fundamentalclassofz. Bypurity[Mi,VI.5.1],Hj2zrj−1(Wk(cid:22);Zl(r))=0. Thus
there is a short exact sequence of G -modules,
k
(1.2.3)
0!H2r−1(Wk(cid:22);Zl(r))!H2r−1((W −jzj)k(cid:22);Zl(r))!Hj2zrj(Wk(cid:22);Zl(r))0 !0:
Applying the (cid:12)rst coboundary map
(cid:14)1:2:3 :Hj2zrj(Wk(cid:22);Zl(r))G0k !H1(Gk;H2r−1(Wk(cid:22);Zl(r)))
to bzc and then taking the image under the tautological map
H1(Gk;H2r−1(Wk(cid:22);Zl(r)))!H1(Gk;H2r−1(Wk(cid:22);Zl(r))=t)’Jlr(W)Gk
gives (cid:11)r (z). De(cid:12)ne
l;k
(1.2.4) arW;l :Zhrom(Wk(cid:22))!Jlr(W); arW;l :=l−i!mk (cid:11)rl;k:
(1.2.5) Proposition. The map (cid:11)r factors through CHr (W). Both (cid:11)r and
l;k hom l;k
ar factor throughCHr (W(cid:22))and arefunctorial with respect tocorrespondences.
W;l hom k
Proof. [Ja, 9.8] or [Sch, 1.10].
For a more detailed discussionofl-adic Abel-Jacobimaps, the readeris referred
to [Ja] and [Sch, x1].
(1.2.6) De(cid:12)nition. Themodl Abel-Jacobimap,denoted(cid:11)(cid:22)r ,isthecomposition
l;k
of (cid:11)r with the tautological map Jr(W)Gk !Jr(W)Gk=l.
l;k l l
1.3. The mod l Abel-Jacobi map for curves. Let C=k be a smooth complete
curve and let c0 be a degree one point. Consider the following diagram:
C(cid:21) −−−(cid:21)i0−! Pic0(C)
? ?
? ?
(1.3.1) ym(cid:21)l yml
C −−−i0−! Pic0(C);
where i0(c) = OC(c−deg(c)c0), ml is multiplication by l, C(cid:21) is the (cid:12)ber product
and(cid:21)i0 andm(cid:21)l arethe canonicalprojections. Fora (cid:12)xeddegreeonepointc distinct
from c0 de(cid:12)ne
(cid:14)1:3:2 :Hf2c;c0g(Ck(cid:22);(cid:22)l)G0k !H1(Gk;H1(Ck(cid:22);(cid:22)l))
to be the (cid:12)rst coboundary map associated to the short exact sequence of G -
k
modules,
(1.3.2) 0!H1(Ck(cid:22);(cid:22)l)!H1((C−fc;c0g)k(cid:22);(cid:22)l)!Hf2c;c0g(Ck(cid:22);(cid:22)l)0 !0:
THE IMAGE OF THE l-ADIC ABEL-JACOBI MAP 801
(1.3.3) Lemma. Suppose that Gk acts trivially on H1(Ck(cid:22);(cid:22)l). Then
(i) Ker(m ) consists of degree one points.
l
(ii) k(C(cid:21))=k(C) is Galois, in fact Abelian.
(iii) There are canonical isomorphisms
H1(Gk;H1(Ck(cid:22);(cid:22)l))’Hom(Gk;H1(Ck(cid:22);(cid:22)l))
’H1(Ck(cid:22);(cid:22)l)’Ker(ml)(k)’Gal(k(C(cid:21))=k(C)):
Proof. (i) follows from the canonical identi(cid:12)cation Ker(ml) ’ H1(Ck(cid:22);(cid:22)l). (ii)
follows from (i) as does (iii) once one notes that the second isomorphism in (iii) is
obtained by evaluating a homomorphism at the Frobenius element (cid:30)2G .
k
(1.3.4) Proposition. (i) The isomorphism (1.1.5)(iii) identifies (cid:14)1:3:2(c−c0) with
the mod l Abel-Jacobi map (cid:11)(cid:22)rl;k for C evaluated at c−c0.
(ii) The isomorphism in (1.3.3)(iii) identifies (cid:14)1:3:2(c −c0) with the Frobenius
element Frob 2Gal(k(C(cid:21))=k(C)).
c
Proof. (i) is straightforward. (ii) is a special case of [Sch, 1.14]. (The sign in loc.
cit. is di(cid:11)erent since Frob in loc. cit. was inadvertently de(cid:12)ned to be the inverse
w
of the usual Frobenius. The sign is irrelevant in the sequel.)
1.4. A strategy for proving surjectivity of the l-adic Abel-Jacobi map.
The following lemma gives a strategy for proving surjectivity of l-adic Abel-Jacobi
maps which involves only computations mod l.
(1.4.1) LSemma. Let k1 (cid:26) k2 (cid:26) k3 (cid:26) ::: be a sequence of finite extensions of k0
such that k =k(cid:22) and the composition
n2N n
(cid:11)(cid:22)rl;kn : Zhrom(Wkn) −−(cid:11)−rl,k−n! Jlr(W)Gkn −−−−! Jlr(W)Gkn=l
is surjective for each n. Then ar :Zr (W(cid:22))!Jr(W) is surjective.
W;l hom k l
Proof. Since Jr(W)Gkn is a (cid:12)nite Abelian l-group,(cid:11)r is surjective if and only if
l l;kn
(cid:11)(cid:22)r is. The lemma follows, since ar is the direct limit of the (cid:11)r ’s.
l;kn W;l l;kn
In practice it is useful to break the l-adic intermediate Jacobian up into pieces
and to prove surjectivity for each individual piece separately using an argument
similarto(1.4.1). Inordertocarryoutthisprogramitisnecessarytodiscusssome
variants of the l-adic Abel-Jacobi map.
2. Variants of the l-adic Abel-Jacobi map
2.1. Cycles supported in (cid:12)bers and the Leray spectral sequence. Let
k0 (cid:26)k be an extension of (cid:12)nite (cid:12)elds. Assume the following
(2.1.1) Geometric situation. f :W !X isaflat,genericallysmoothmorphism
ofsmoothpropervarietiesoverk0. X isacurve,W hasdimension2m+1(cid:21)3,and
the geometric (cid:12)bers of f are connected. The inclusion of the largest open subset
over which f is smooth is denoted j : X_ ! X. The inclusion of the generic point
is denoted g : (cid:17) ! X. After base changing to k we have a diagram in which all
802 CHAD SCHOEN
squares are Cartesian:
W0 −−−−! W −−−− W_ −i−V−− V
? ?k ?k ?
? ? ? ?
f0y fy f_y y
X0 −−−−! X −−j−− X_ −i−x−− x
k k
Here i is a k-rationalpoint and X0 is the complement of the image of x in X .
x k
Given the geometric situation (2.1.1) and a cycle z 2 Zm+1(W ) supported on
hom k
V there is a restriction map from (1.2.3) (with r =m+1) to
(2.1.2) 0! iV(cid:3)H(H2mV2+m1+(1W(Wk(cid:22);k(cid:22)Z;lZ(ml(m++1)1)))) !H2m+1(Wk(cid:22)0;Zl(m+1))
!HV2m+2(Wk(cid:22);Zl(m+1))0 !0:
Let L(cid:15) denote the (cid:12)ltration on H2m+1(Wk(cid:22);Zl(m+1)) and (L0)(cid:15) the (cid:12)ltration
on H2m+1(Wk(cid:22)0;Zl(m+1))) resulting from the Leray spectral sequence for f (re-
spectively f0). From the commutative diagram
HV2m+1(Wk(cid:22)x;Zl(m+1)) −−i−V−(cid:3)! H2m+1(Wk(cid:22)x;Zl(m+1))
? ?
(2.1.3) ? ?
Hx2(Xk(cid:22);R2m−1f(cid:3)Zl(m+1)) −−−−! H2(Xk(cid:22);R2m−1f(cid:3)Zl(m+1));
in which the left-hand arrow is an isomorphism by purity and the bottom arrow is
surjective, it follows that
(2.1.4) iV(cid:3)(HV2m+1(Wk(cid:22);Zl(m+1)))=L2:
Thus
(2.1.5) 0!L1=L2 !(L0)1 !HV2m+2(Wk(cid:22);Zl(m+1))0 !0
may be identi(cid:12)ed with an exact subsequence of (2.1.2). We may rewrite (2.1.5) in
terms of the cohomology of H:=R2mf(cid:3)Zl(m+1) on the curve X. In the spectral
sequences for f and f0, E1;1 = E1;1. Also (L0)2 = 0. One deduces that (2.1.5) is
2 1
isomorphic to
(2.1.6) 0!H1(Xk(cid:22);H)!H1(Xk(cid:22)0;H)!Hx2(Xk(cid:22);H)0 !0:
Write bzcV for the image of bzc in HV2m+2(Wk(cid:22);Zl(m + 1))G0k and bzcx for the
image of bzcV in Hx2(Xk(cid:22);H)G0k. Throughout this paper (cid:14)(cid:3) will denote the (cid:12)rst
coboundarymaponG -cohomologyassociatedtoashortexactsequence(*). With
k
this notation (cid:14)2:1:5(bzcV) maps to (cid:14)2:1:6(bzcx). Since (cid:14)1:2:3(bzc) (with r = m+1)
mapsto(cid:14)2:1:5(bzcV),weconcludethatthecoboundarymap(cid:14)2:1:6 givesinformation
concerningthevalueofthel-adicAbel-Jacobimapatacyclesupportedinasmooth
(cid:12)ber of f.
2.2. Cycles supported in (cid:12)bers and a criterion for surjectivity of the
l-adic Abel-Jacobi map. Assume the geometric situation (2.1.1). Denote by
THE IMAGE OF THE l-ADIC ABEL-JACOBI MAP 803
Zm(f−1(x))0 the inverse image of Zhmo+m1(Wk) under the canonical injection
Zm(f−1(x))!Zm+1(W ). De(cid:12)ne
k
M
Zfm+1(Wk):=im[ Zm(f−1(x))0 !Zhmo+m1(Wk)]
x2X_k(k)
and Zfm+1(Wk(cid:22)) = l−i!mZfm+1(Wk), where the limit is over (cid:12)nite intermediate (cid:12)elds
k0 (cid:26) k (cid:26) k(cid:22). Clearly Zfm+1(Wk(cid:22)) may be regarded as the subgroup of Zhmo+m1(Wk(cid:22))
generated by cycles supported on the smooth (cid:12)bers. There is a canonical map
(2.2.1) af :Zfm+1(Wk(cid:22))!J(L1=L2)
whose value at a cycle z is computed as follows: First note that z is de(cid:12)ned over
a (cid:12)nite extension k (cid:27) k0 and is supported in a (cid:12)ber V. Thus z 2 Zm(Vk)0. Then
af(z) is given by the image of (cid:14)2:1:5(bzcV) 2 H1(Gk;L1=L2) in J(L1=L2)Gk. The
image of a (z) under the natural map J(L1=L2) !(cid:16)1 J(L0=L2) coincides with the
f
image of am+1(z) under the natural map Jm+1(W)!(cid:16)2 J(L0=L2).
W;l l
Write LiJm+1(W) for the image of J(Li) in Jm+1(W).
l l
(2.2.2) Lemma. am+1(Zm+1(W(cid:22)))(cid:26)L1Jm+1(W).
W;l f k l
Proof. By (1.1.3)(ii) J is a right exact functor. Thus the rows in the commutative
diagram
0 −−−−! L1Jm+1(W) −−−−! Jm+1(W) −−−−! J(L0=L1) −−−−! 0
l l ? (cid:13)
? (cid:13)
y (cid:13)
(cid:16)2
J(L1=L2) −−−(cid:16)1−! J(L0=L2) −−−−! J(L0=L1) −−−−! 0
are exact. One deduces that L1Jlm+1(W) = (cid:16)2−1((cid:16)1(J(L1=L2))) and the assertion
follows.
(2.2.3) Proposition. In order to show that am+1 : Zm+1(W(cid:22)) ! Jm+1(W) is
W;l hom k l
surjective it suffices to verify that the following three conditions hold:
(i) am :Zm (V(cid:22))!Jm(V) is surjective.
V;l hom k l
(ii) a is surjective.
f
(iii) The composition of am+1 with the tautological map Jm+1(W)!t J(L0=L1)
W;l l
is surjective.
Proof. Assume (i). The canonical isomorphism
H2m−1(Vk(cid:22);Zl(m))’HV2m+1(Wk(cid:22);Zl(m+1))
and(2.1.4)implythatL2Jlm+1(W)isintheimageofamW+;l1. SinceiV(cid:3)(Zhmom(Vk(cid:22)))(cid:26)
Zm+1(W(cid:22)), this shows that L2Jm+1(W)(cid:26)am+1(Zm+1(W(cid:22))). Assume (ii). Then
f k l W;l f k
(cid:16)2(amW+;l1(Zfm+1(Wk(cid:22))))=(cid:16)1(J(L1=L2)):
SinceL2Jlm+1(W)=Ker((cid:16)2),(i)andtheproofof(2.2.2)implyamW+;l1(Zfm+1(Wk(cid:22)))=
L1Jm+1(W). Finally assume (iii). Then the surjectivity of am+1 follows from
l W;l
L1Jm+1(W)=Ker(t).
l
804 CHAD SCHOEN
2.3. A variant of the l-adic Abel-Jacobi map associated to a subsheaf.
Keep the notations of 2.2. In practice it often happens that the inverse system
H(−1) = fR2mf(cid:3)Z=ln(m)gn2N contains a direct factor E := fEngn2N which satis-
(cid:12)es:
(i) 8 x2X_(k(cid:22)); Ex :=l im−(En)x is torsion free, and
(2.3.1) (ii) Hx2(Xk(cid:22);E(1))=Hx2(Xk(cid:22);E(1))0:
It is in this context that we wish to de(cid:12)ne the l-adic Abel-Jacobi map on Tate
cycles. One application will be to the situation where f comes from a Lefschetz
pencil of hyperplane sections and E comes fromthe vanishing cohomology(cf. x9).
For each x2X_(k(cid:22)) an open subgroupof Gk acts on Ex ’Hx2(Xk(cid:22);E(1)). Write g
for the l-adic Lie algebra of the image. TLhe subgroup annihilated by g, Exg, is the
subgroup of Tate cycles. De(cid:12)ne Z(E) = x2X_(k(cid:22))Exg. There is an abstract l-adic
Abel-Jacobi map
(2.3.2) aE :Z(E)!J(H1(Xk(cid:22);E(1)));
whichmaybeevaluatedonaTateclassz 2Eg asfollows: Choosea(cid:12)niteextension
x
(cid:12)eldk (cid:27)k0 suchthatz 2ExGk andlet(cid:14)2:3:3 bethe(cid:12)rstcoboundarymapassociated
to the short exact sequence of G -modules
k
(2.3.3) 0!H1(Xk(cid:22);E(1))!H1(Xk(cid:22)0;E(1))!Hx2(Xk(cid:22);E(1))0 !0:
Now aE(z) is de(cid:12)ned to be the image of (cid:14)2:3:3(z) under the natural map
H1(Gk;H1(Xk(cid:22);E(1)))!J(H1(Xk(cid:22);E(1))):
It is clear that aE(z) does not depend on the choice of k.
A choice of projection q :H(−1)!E gives rise to a map
q#x : Zm(f−1(x))0 !Hf2−m1+(x2)(Wk(cid:22);Zl(m+1))g ’Hx(−1)g !qx Exg;
and hence to a map
M
(2.3.4) q# := q#x :Zfm+1(Wk(cid:22))!Z(E):
x2X_(k(cid:22))
There is also a map
q(cid:3) :J(L1=L2)’J(H1(Xk(cid:22);H)) −J−(−H−1−(q−!)) J(H1(Xk(cid:22);E(1))):
(2.3.5) Lemma. q(cid:3)(cid:14)af =aE (cid:14)q#.
Proof. Straightforward.
2.4. A criterion for the surjectivity of aE using only computations mod l.
We (cid:12)rst introduce a mod l version of the map aE, which will be easier to compute
thanaE itself. SupposethattheinversesystemfEngn isaconstructiblel-adicsheaf
in the sense of [Mi, p. 163] and that E is flat over Z=ln for all n.
n
(2.4.1) Lemma. (i) Suppose that H0(Xk(cid:22);E1(1)) = 0. Then H1(Xk(cid:22);E(1)) is tor-
sion free.
(ii) Suppose that j(cid:3)En is locally constant for all n and H0(X_k(cid:22);E1_) = 0. Then
H1(Xk(cid:22);E(1))=l’H1(Xk(cid:22);E1(1)).
(iii) If (i) and (ii) hold, then
J(H1(Xk(cid:22);E(1)))Gk=l’H1(Gk;H1(Xk(cid:22);E1(1))):