Motivic Sheaves and Filtrations on Chow Groups

Proceedings of Symposia in Pure Mathematics Volume 55 (1994), Part 1

Motivic Sheaves and Filtrations on Chow Groups

U W E J A N N S E N

Grothendieck's motives, as described i n [Dem, K12, M a ] are designed as a tool to understand the cohomology o f smooth projective varieties a n d the algebraic cycles modulo homological and numerical equivalence o n them.

A c c o r d i n g to Beilinson and Deligne, Grothendieck's category o f pure motives should embed i n a bigger category o f m i x e d motives that allows the treatment o f arbitrary varieties and an understanding o f the whole C h o w group o f cycles modulo rational equivalence, i n fact, even o f all algebraic .fiT-groups o f the varieties.

In this paper we review some o f these ideas and discuss some conse- quences. In particular, we show how the vast conjectural framework set up by Beilinson leads to very explicit conjectures on the existence o f certain f i l i a - tions on C h o w groups o f smooth projective varieties. These filtrations w o u l d offer an understanding o f several phenomena and counterexamples that for some time have led people to believe that the behaviour o f the algebraic cycles is absolute chaos for codimension bigger than one.

In § 1 we review some basic facts o n C h o w groups, correspondences, and cycle maps into cohomology theories. We recall a counterexample o f M u m - ford i m p l y i n g that i n general C h o w groups are not representable a n d the Abel-Jacobi map has a huge kernel and some investigations o f B l o c h o n this topic.

In §§2 and 4 we state altogether four versions o f Beilinson's conjectures on m i x e d motives and filtrations on C h o w groups, increasing i n generality and sophistication. The first one does not even mention m i x e d motives and proposes finite filtrations F⁰ D F¹ D • • • on rational C h o w groups CH^j (X)^Q that are uniquely determined by their behaviour under algebraic correspon- dences. The first step is homological equivalence, but the following steps differ very m u c h from those considered classically. F o r example, algebraic equivalence does not appear, and the second step is something like the kernel

1991 Mathematics Subject Classification. Primary 14C15; Secondary 14A20, 14C25.

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© 1994 American Mathematical Society 0082-0717/94 $1.00+ $.25 per page

of (all) Abel-Jacobi maps. T h e crucial property is that the actions o f corre- spondences o n the graded pieces GT^UFCH^J(X)^Q factor through homological equivalence. T h i s w o u l d very nicely bridge the gap between homological and rational equivalence, i n finitely many steps.

The described "filtration conjecture" can thus be regarded as a counterpart of the standard conjectures, being responsible for the part not covered by the latter. It is amazing to realize that the filtration conjecture would follow from the injectivity o f suitable cycle maps, while (parts of) the standard conjectures w o u l d follow from the surjectivity o f cycle maps, v i z . , the conjectures o f Hodge and Tate.

W h i l e by definition GV^0FCH^J(X)^Q is the space o f algebraic cycles i n the cohomology H^2J(X) o f degree 2j, the higher graded pieces are related to H^2J~^L(X), H^2J~²(X), and so on. In fact, version 2 o f the conjecture pro- poses the description (called "Beilinson's f o r m u l a " i n the following) (0) G/^FCH^J(X)^Q = E x t ^ ( l , h^2j~^v(X)(J)),

in terms o f Y o n e d a extensions i n the conjectural category JtJK^k o f m i x e d motives, where 1 is the t r i v i a l motive and h^L(X) is the pure motive corre- sponding to the /th cohomology o f X. T h i s is closely related to higher cycle maps, a n d indeed M u m f o r d ' s counterexample and the above isomorphisms suggest studying higher than secondary (=Abel-Jacobi) maps to understand the whole C h o w group.

V e r s i o n 3 o f the conjecture reveals more o f the framework o f m i x e d mo- tives. R e c a l l that X (&h^L(X) is the universal W e i l cohomology theory for smooth projective varieties X and that by definition an element a e CH^J(X)^Q corresponds to a m o r p h i s m g>^a: 1 -^h^LJ(X)(J) i n Grothendieck's category Jt^k o f pure motives. Roughly speaking, Beilinson's f o r m u l a should come f r o m a derived version: I f one believes that ti (X) arises as the coho- mology o f a complex R(X) i n D^b(JKJ?'^k), the bounded derived category o f JKJK^k , and the m o r p h i s m (p^a from a m o r p h i s m t]^a: 1 -> R(X)(j)[2j] (such that cp^a is obtained by passing to the Oth cohomology), then this leads to a cycle map

(1) CH^j(X)^q - H o n ^{v w} O , R(X)(j)[2j]) and to an induced nitration v i a the spectral sequence

(2) E x t ^^{f t}( 1 , h^q{X){j)) H o n v ^ U , R(X)(j)[p + q]).

The optimistic conjecture says that (1) is an isomorphism. T h i s together with the degeneration o f (2) w o u l d give Beilinson's f o r m u l a (0).

A l l this only reflects the situation encountered i n the £-adic cohomology, and i n fact, (2) is the m o t i v i c analogue o f the Hochschild-Serre spectral se- quence a n d (1) is the analogue o f the cycle map into ^-adic cohomology over k. T h e most general version 4 o f Beilinson's conjecture expresses the hope

that there is a theory o f m i x e d motives that resembles and parallels closely the general theory o f ^-adic cohomology and ^-adic sheaves. In particular, one hopes for relative and local versions o f m i x e d motives, the so-called m i x e d m o t i v i c sheaves.

In §3 we state some remarkable consequences o f Beilinson's conjectures w h i c h already follow from the down-to-earth version 1, but which we derive v i a Beilinson's formula, to demonstate its use and usefulness. In particular, we show that the conjectures w o u l d lead to a good understanding o f the representability o f C h o w groups. G u i d e d by the conjectural theory, we prove some results on the nonrepresentability o f C h o w groups. T h i s extends work o f M u m f o r d , B l o c h , R o i t m a n , and others and may be interesting i n its o w n right.

F i n a l l y , i n §5 we discuss the relationship with a conjectural filtration pro- posed by M u r r e . T h i s proposal has the advantage o f being quite explicit, i n the formulation close to the standard conjectures, and more amenable to being proved i n part. Whereas M u r r e arrived at this conjecture by the consid- eration o f decompositions o f motives, it was quickly clear to several experts that it is i m p l i e d by Beilinson's conjectures. A t the Seattle "motives" con- ference I discussed a partial converse, and soon after it occurred to me that M u r r e ' s conjecture is i n fact equivalent to version 1 o f Beilinson's conjecture.

T h i s paper would not be complete without mentioning that Grothendieck certainly envisioned a much more general theory o f motives than just for smooth projective varieties over a field. In particular, he already thought about motives over arbitrary bases and a general m o t i v i c duality f o r m a l i s m . T h i s becomes quite clear i n a letter written by Grothendieck to Illusie i n 1973 which is reproduced i n an appendix. T h i s letter also addresses several interesting questions on motives that I have not seen discussed elsewhere.

I a m indebted to L . IUusie and W . Messing for p r o v i d i n g me w i t h the letter i n the appendix and other " h i s t o r i c a l " material on motives, and to S. K l e i m a n for numerous helpful comments o n this paper. The m a i n result o f §5 was obtained during a visit o f L e i d e n University, which I thank for its hospitality. It is a pleasure to thank J . P . M u r r e and S. Saito for stimulating discussions. M y stay at the Seattle conference was partially supported by the D F G , whose support is gratefully acknowledged.

1. Higher cycle maps and Bloch's conjecture

1.1. Let X be a smooth projective variety over a field k . F o r any integer j > 0 , the set X^ = {x e JSfIcodim x = j} o f points o f codimension j can be identified with the set o f closed irreducible subvarieties Z o f codimension j i n X (by mapping x to its closure Z = {x} a n d Z to its generic point).

Recall that the group o f cycles o f codimension j on X is the free abelian group Z^j(X) = Q^xex(J)Z on X^U) a n d that the j t h C h o w group CH^j(X) is the quotient o f Z^j(X) modulo cycles that are rationally equivalent to zero

[K12, §2]. B y Q u i l l e n [Qui, §7, proof o f 5.14], we can write this as

(1.1) CH^j(X) = Coker j 0 k(x)^x ^ 0

where k(x) is the residue field o f x (= the function field o f Z = {x}) and d i v is the d i v i s o r map (cf. also [Fu, 1.3 and 1.6] for C h o w groups that are graded by dimension).

T h e following operations on cycles are (only) well defined m o d u l o rational equivalence. O n e has an intersection product

CHⁱ(X) x CH^j(X) -> CH^i+j(X), (a⁹0)~a-p⁹

m a k i n g CH* (X) into a commutative ring, and for a m o r p h i s m / : X —• Y o f smooth projective varieties one has pull-back maps

f* : CH^j(Y) ^ CH^j(X)^y 7 > 0 ,

i n d u c i n g a ring m o r p h i s m CH*(Y) —• CH*(X), and a push-forward map CH*(X) CH*(Y)

m a p p i n g CH^d+i(X) to CH^e^ⁱ(Y)^yH X and Y are o f pure dimensions d and e , respectively. These operations enjoy the following compatibilities:

(a) (functoriality) (gf)* = f*g* and (gf)+ = g+f+ for a second mor- p h i s m g: 7 —• Z .

(b) (projection formula) f^a • /? = / J a • / * / ? ) .

(c) (base change) F o r a Cartesian diagram o f projections

XxYxZ - ^ U ^ x F

F x Z — one has p*q^ = (P^yz)*P*XY •

These properties ensure that one has a bilinear, associative composition law o f correspondences (= algebraic cycles on products)

(1.2)

CH^e^^s (X xYxZ)x CH^d^^r (X x Y x Z) A CH^d+r+e+s(X x Y x Z)

PXY (Pxz)*

CH^e+s(Y xZ) x CH^d+r(X xY) -+ CH^d+r+s(X x Z) (g, f) " g°f=(P^Xz)*(PYz8^mP*XYf)

for X and Y o f pure dimensions d and e , respectively, m a k i n g the diagram commutative. I n particular, CH^d (X x X) is an associative ring w i t h unit

(consider X = Y = Z ; the unit is the class o f the diagonal A : X ^ X x X), and one has an action o f correspondences on C h o w groups (consider X = Spec/c) :

CH*^+S(Y x Z ) x CH^r(Y) -+ CH^r+s(Z), (a, y) ^ ay = (p^z)^(a >p*^Yy).

1.2. M o r e generally, algebraic correspondences act o n generalized coho- mology theories. These are contravariant functors

H:X^H*(X⁹*)

from the category ^cV^k o f smooth projective varieties over k to the category o f bi-graded i?-algebras, for a ring R, equipped w i t h the following additional structure:

(I) F o r every m o r p h i s m / : X -» Y i n ^cV^k there is a map o f i?-modules l:H*(X,*) -> H*(Y,*)⁹ mapping H^2d+i(X⁹d + j) to H^2e+i(Y⁹e + j) i f X a n d Y are o f pure dimensions d and e, respectively, such that the properties (a)-(c) h o l d correspondingly, where / * = H(f).

(II) There are cycle maps

cl: CH^j(X)^H^2j(X⁹J) compatible w i t h pull-back, push-forward, and products.

In fact, by (I) one has analogues o f the diagram (1.2) and, hence, com- position and action o f "cohomological" correspondences, i.e., elements i n H*(X x Y, * ) . B y (II), algebraic correspondences act v i a their cycle classes.

E x p l i c i t l y , we have the following diagram, for X o f pure d i m e n s i o n d , (1.3)

H^2d+2r(X xY⁹d + r) x Hⁱ (X x Y, m) -> H^2d+i+2r(X xY⁹d + m + r)

Cl Px (JPY

CHd+r(X xY) x Hi(X, m) - Hi+2r(Y, m + r).

1.3. Examples o f generalized cohomology theories are:

(1) T h e C h o w theory:

[ 0 otherwise, where R = Z and cl is the identity.

(2) Every W e i l cohomology theory X *-+ H*(X) (cf. [K13, Chapter 3]) Hⁱ(X⁹J) = Hⁱ(X) for a l l j.

(3) Singular cohomology with Hodge-Tate twists

Hⁱ(X, j) = Hⁱ(X(C), Z(J)), Z(J) = Z(2n^\)^j ,

for k = C⁹ where R = Z and the cycle class is renormalized (cf. the discus- sion i n [Ja3, §5]);

(4) Gaelic cohomology w i t h Tate twists

Hⁱ (X, j) = H^ietQC⁹ Z¹U)), X = X x^kI ⁹

for £ / char(/c), where R = Zⁱ. Here k is a separable closure o f k and Z^/( y ) = Z^t(\)®^J, where Z^(I) = I i m fi^tn for the sheaves o f ^ " t h roots o f unity.

(5) Deligne cohomology [ B e l ; E V⁵ §1]

Hⁱ(X⁹J) = H^(X⁹ZU)) for k = C.

(6) ^-adic cohomology over k

Hⁱ(X⁹J) = Hl^x(X⁹Z^lU)) (naive or continuous [Jal]) for I ^ char(A:).

After tensoring w i t h Q , examples (3) and (4) are also examples (the p r i n c i p a l ones) o f W e i l cohomology theories, but we have some a d d i t i o n a l structure: In example (3) the cohomology group H¹(X, j) carries a pure Hodge structure o f weight i - 2j, by identifying it w i t h the tensor product o f H¹ (X(C), Z), w h i c h has the usual Hodge structure o f weight / , a n d the Hodge structure Z(j), w h i c h has weight -2j (cf. [De2, 2.1 a n d 2.2]). T h e n cup-product, pull-backs, and push-forwards are compatible w i t h the Hodge structures, and the fact that the cycle map has image i n

H^2j(X(C)⁹Z(J))HH^jj(X)

implies that the action o f correspondences respects the Hodge structures. T h e last statement becomes quite obvious, i f one identifies the above group w i t h

H o m^{H S}( Z , / /^{2 y}' ( X ( C ) , Z a ) ) ) ,

the group o f homomorphisms o f Hodge structures, where Z is the t r i v i a l Hodge structure. In example (4), the absolute G a l o i s group G^k = Gal(k/k) o f k acts continuously o n H¹ (X, j) by functoriality o f etale cohomology.

Since the cycle map has image i n the fixed module

H^2j (X⁹ Z,(j)f^k S H o m ^ (Zⁱ, H^lj (X⁹ Zⁱ(J))),

the action o f correspondences respects the G a l o i s actions. W e remark that H¹ (X, j) is again pure o f weight i -2j as a G a l o i s module, i.e., Zⁱ-sheaf on ( S p e c f c )^a (cf. [De3, 3.4.11]).

1.4. Although the cycle maps i n (3) and (4) and, more generally, the cycle maps into W e i l cohomology theories are already objects o f interesting study and deep conjectures, namely, the Hodge conjecture, the Tate conjecture [Ta], and the standard conjectures [K13], this only covers a small part o f the C h o w groups, namely, cycles m o d u l o homological equivalence. R e c a l l that a cycle a e Z^j(X) is called:

(i) homologically equivalent to zero (a ~^hom 0 ) , i f cl(a) = 0 for a cycle m a p cl into a W e i l cohomology, and

(ii) numerically equivalent to zero (a ~^num 0 ) , i f the intersection number (a • fi) = d e g a • p = 0 for a l l P e CH^d~^j(X), i f X is o f pure dimension d , say.

F r o m these definitions it is only clear that a ~^hom 0 implies a ~^num 0 , a n d a p r i o r i ~^hom depends o n the chosen W e i l cohomology, but according to the standard conjectures ~^hom and ~^num should coincide, w h i c h w o u l d i m p l y the independence as well. N o w CH^j(X)/ ~^num is a finitely generated abelian group (cf. [K13, 5-2]), while

CH^j(X)^hm = CH^J(X)\cl(a) = 0}

can be huge. T o a certain extent this group can be studied by secondary cycle maps, namely, the Abel-Jacobi maps.

F o r k = C this is a map

'° H^2j-¹ (X(C)⁹ Z(J)) + F^j

i n t o the Weil-Griffiths intermediate Jacobian (cf. [Lie, p. 131]). Here F^j =

®^p+^q=lj-^X,^p>jH^p,q(X) is the jih step o f the Hodge filtration, CH^j(X)⁰ is

the kernel o f the cycle map cl i n example 1.3(3), and we have n o r m a l i z e d the lattice to be H^2j~^l (X, Z(j)) instead o f the more classical H^2j~^x(X, Z).

N o t e t h a t CH^j(X)⁰ is o f finite index i n CH^j(X)^hom . B y t h e c l a s s i c a l A b e l - Jacobi theorem, ct is an i s o m o r p h i s m for j = I.

The ^-adic analogue, for £ ^ char(/c), is a map

cl': CH^j(X)⁰ -> H¹ (G^k9 H^2j'¹ (X⁹ Zⁱ(J))),

w h i c h can be defined by the cycle map into H^2j(X⁹ Zⁱ (j)) and the H o c h s c h i l d - Serre spectral sequence

(1.4) E^pS = H^p(G^k , H^qCX⁹ Zⁱ(J))) =• H^p^^q(X⁹ Zⁱ(J))

(cf. [ J a l , 6.15c)]). Here CH^j(X)⁰ is the kernel o f the cycle map into H^2j(X⁹ Zⁱ(J)); it is o f finite index i n

CHⁱ(X)^jum = Kcr(CH^j(X) - H^2j(X⁹ Qⁱ(J))).

F o r j = 1 and a finitely generated field k, ct is injective up to torsion prime to £ by K u m m e r theory and the M o r d e l l - W e i l theorem (cf. [ J a l , 6.15a], where "up to torsion p r i m e to I " should be added).

In the ^-adic setting we have i n a d d i t i o n higher than secondary cycle classes. In fact, let F⁰ D F¹ D • • • be the filtration o n H^2j(X, Qⁱ(J)) (con- tinuous etale cohomology over k) induced by the Hochschild-Serre spectral sequence

(1.5) E^{p2 q} = H^p(G^k , H^qCX⁹ Qⁱ(J))) H^p+⁹(X⁹ Qⁱ(J)).

Since this spectral sequence degenerates (cf. the remark i n [Jal, 6.15b]; this fact also follows from the considerations i n [De5]), we have isomorphisms

GT^UFH^2j(X, Q¹(J)) * H^U(G^K , H^V~^V(X, Q¹(J)).

N o w let F¹ be the descending filtration o n CH^j(X) obtained by pull-back v i a the cycle map

cl: CH^j(X)-, H^2J(X, Q^e(J)).

Hence i f = CH^j(X), F^xt = CH^j(X)^hom, and F² = kernel o f the A b e l - Jacobi map cl', up to torsion. T h e n we obtain higher cycle maps

(1.6) cl^[v): F^UICH^j(X) - H^U(G^K , H^2j~^u(X, Q¹(J)))

w i t h kernel F^U^^RLCH^j(X) extending the cases v = O a n d 1 discussed before.

A s we remarked, CH¹(X)^hom can effectively be recovered by Abel-Jacobi maps. B u t i n any case, the structure o f CH^j(X) is well understood for j = 1 : One has a canonical i s o m o r p h i s m CH¹ (X) = P i c ( X ) and an exact sequence

O P i c⁰( Z ) -+ P i c ( X ) -+ NS(Xfk) -+ 0 ,

where the " ^-rational Neron-Severi group" NS(Xfk) is a finitely generated abelian group and P i c⁰( X ) is the group o f points o f an abelian variety—

namely, the Jacobian variety J a c ( X ) — i f X has a rational point. In fact, J a c ( X ) = ( P i c⁰^ )^{r e d} for the P i c a r d group scheme P i c ^ (cf. [Gr2, 3.2]), there is an exact sequence O —• P i c ( X ) Pic^x^^k(k) —> Br(k) i n w h i c h j is an i s o m o r p h i s m for X(Zc) ^ 0 (cf. [Grl, 2.1]), and we obtain the result by putting P i c⁰( X ) = T¹P I C^{0 y a}( Z C ) , since NS(Xfk) is finitely generated for algebraically closed k (cf. [SGA6, X I I I , 5.1]). It is k n o w n that P i c⁰( X ) has finite index i n CH¹(X) .

v 'num

The naive hope that similar structure results w o u l d h o l d also for j > 2 was destroyed by the following counterexample o f M u m f o r d . F o r X o f d i - mension d over k let CH^d(X)⁰ be the C h o w group o f zero cycles o f degree zero on X . I f k = C (or a universal domain), then, following M u m f o r d , call CH^d(X)⁰ finite dimensional, i f there is an n e N such that the natural map

( 1 ?) SⁿXik) x SⁿX(k) -> CH^d(X)⁰, ( a , b) —• class o f a - b is surjective, where SⁿX is the « t h symmetric power o f X .

THEOREM

1.5 [Muml, p. 203]. Let X be a smooth projective surface over C If H²(X⁹^^x) ^ 0 , then CH²(X)⁰ is not finite dimensional

For the discussion o f this result it is useful to investigate various possible characterizations o f a "nice behaviour" o f CH^d(X)⁰ .

PROPOSITION

1.6. Let X be a smooth, projective, geometrically irreducible variety of dimension d over a field k, and let Q D k be an algebraically closed field. Consider the following statements.

(i) (resp. ( i ' ) ) There is an n eN such that the natural map SⁿX(Q) x SⁿX(Q) -4 CH^d(X^a)⁰

(resp. SⁿX(Q) x SⁿX(Q) -+ CH^d(Xⁿ)⁰ ® Q) is surjective, where X^a = Xx^k Q.

(ii) (resp. (ii')) There exists a smooth projective curve C over Q and a morphism f: C -> X^a such that

P i c⁰( C ) = CH¹(C)⁰ - CH^d(X^a)⁰ (resp. X : CHⁱ (C)⁰ <g> Q -> CH^d(X^a)⁰ ® Q) is surjective.

(iii) (resp. (iii')) There exists a closed subscheme Y C^zX of dimension 1 such that

CH^d((X-Y)^a) = O

(resp. CH^d ((X - Y)^a)^Q = 0 ) , where CH^j(Z) is defined for j>0 and any algebraic Q-scheme Z by formula (1.1), applied to Z over Q in place of X over k.

(iv) (resp. (iv')) If B C X^a is a smooth linear space section of dimension 1, then

CH^d(X^a-B) = O (resp. CH^d(X^a-B)^q = O).

(v) (resp. ( v ) ) The canonical map

U^Xq: CH^d(X^a)⁰ - A l b ( X ) ( Q )

is an isomorphism (resp. an isomorphism after tensoring with Q ) , where A l b ( X ) is the Albanese variety of X.

Then (v) (v') => (iv) (iv') => (iii) (iii) (ii) & (ii') (i) & (i).

If Q is uncountable, then (i) => ( v ) , so all statements are equivalent. Property (v) holds over an algebraically closed field QDk if and only if it holds for all algebraically closed fields Q^f, Q D Q' D k, of finite transcendence degree over k. In particular, (v) holds for all algebraically closed fields QD k, if it holds for one which is uncountable.

PROOF,

(V) =» (iv) L e t B ^ X^a be as i n (iv), and consider the following commutative diagram.

a^B: CH¹(B)⁰ A l b ( U ) ( O )

- I I--

a^Xa: CH^d(X^il)⁰ • Mb(X^a)(O) = Mb(X)(Q)

The right-hand map is surjective by the choice o f B (cf. [Wei, C o r o l - lary 1 to Theorem 7]). T h e canonical map a^B is an isomorphism, since Jac(C) = A l b ( C ) for every smooth projective curve C. Hence (v) implies that the left-hand map ^ is surjective as well, and then the same is true for i^m: CH^x(B) CH^d(X^a), since B has a point o f degree 1. N o w (iv) follows from the exact sequence

CH^x(B) CH^d(X^a) -+ CH^d(X^a - B ) ^ 0 (cf. 3.10(2)).

(iv) (iii) There always exists a linear section B as i n (iv) that is defined over a finite separable extension o f k, and we may take Y = image o f B i n X (which is not necessarily smooth). Since B is a closed subvariety o f

Y^a, the natural restriction CH^d(X^a -B)-^ CH^d(X^a - Y^a) (cf. 3.10(1)) is surjective; hence the i m p l i c a t i o n .

(iii) =>• (ii) W e may assume that Y is o f pure d i m e n s i o n 1 and then take C = normalization o f ( I^{r n})^{r e d} . In fact, (iii) implies that the first map i n the exact sequence

CH¹(Y^a) - CH^d(X^a) -* CH^d(X^a - Y^a) - 0

(cf. 3.10(2)) is surjective, but it is easy to see that the map f^% i n (ii) induced by / : C - (Y^a)^red -+Y^a-^X^a factors as

CH^x(C) - CH^x((Y^a)^red) = CH^x(Y^a) - CH^x(X^a), where the first map is induced by C^{( 1 )} —• Y^a\ x ^ f(x) and, hence, is surjective.

(ii) => (i) If C is an irreducible smooth projective curve o f genus g over Q, then it follows from the R i e m a n n - R o c h theorem that S^gC(Q) x S⁸C(Q) - CH^x(C)⁰ is surjective (note that SⁿC(Q) is identified w i t h the set o f effective divisors o f degree n). Hence the c l a i m follows from the c o m - mutative diagram

SⁿX(Q) x SⁿX(Q) -> CH^d(X^a)⁰ T / . T / .

SⁿC(Q) x SⁿC(Q) -» CH^x(C)⁰

That (i) implies (v), for an uncountable field Q, is a theorem o f R o i t m a n ([Roil, Theorem 4, p. 585]; i n R o i t m a n ' s paper it is generally assumed that char(Q) = 0 , but the proof o f this result does not involve this assumption).

B y another theorem o f R o i t m a n [Roi2, 3.1], a^Y always induces an iso- morphism

(1.8) T o r ( C Z ^ ( Xⁿ)⁰) - T o r ( A l b ( J f ) ( Q ) )

on the torsion subgroups (for all smooth projective varieties X o f dimension d over k and QDk algebraically closed). T h i s i m p l i e s the equivalence o f

(v) and ( v ' ) , since a^Y is always surjective. Another consequence is that i , : T o r ( C Z Z¹( U )⁰) - » T o r ( C Z Z ^ ( Xⁿ)⁰)

is surjective for any smooth linear space section o f dimension 1, / : B <-* X ⁿ , by the surjectivity o f Z^{i l i}: A l b ( U ) -> A l b ( Xⁿ) . Since CH^d(X^a)⁰ is divisible, this implies the equivalence o f (iv) and ( i v ' ) , and the equivalence o f (iii) and ( i i i⁷) , (ii) and ( i i ' ) , or (i) and (i') follows as well, using the ingredients o f the proofs o f the various implications.

F i n a l l y , i f QD Q' are algebraically closed extensions o f k and i f a^Y <g>Q

Aft is injective, then so is a^{x ;} <g> Q , since the restriction map CZZ (X^a>)^q —•

CH^d(X^a)^q is injective (cf. 3.10(4)). Conversely, since CH^d(X^a) = Iim CH^d(X^at)^y

where the l i m i t is over all algebraically closed fields Q' ⁹ Q D Q' D k, o f finite transcendence degree over k (cf. 3.10(3)), a^y is injective i f a^y is

An An '

injective for a l l such Q'. T h i s shows the remaining claims. F o r the last c l a i m note that X can be defined over a field k^Q w h i c h is finitely generated over the p r i m e field a n d that an uncountable algebraically closed field QDk contains all fields Q^f D k⁰ o f finite transcendence degree over k⁰.

REMARK

1.7. C o n d i t i o n 1.6(iii) and its generalization to d i m Y > 1 ap- pears i n work o f B l o c h and Srinivas [BS] and is further investigated i n [MS]

and [SaS] (cf. also §3). C o n d i t i o n 1.6(ii) appears i n [Bll, lecture 1, A p - pendix], and the equivalence o f (iii) and (v) is stated without proof i n [MS,

1.2(b)]. F o l l o w i n g B l o c h (cf. [Bll, D e f i n i t i o n (1.1); BS, p. 1238]), we shall call CH^d(X)ⁿ representable, i f a^y is an isomorphism over a universal do- m a i n QDk.

I f k - C , then there is an i s o m o r p h i s m

A l w y w n ~ H^ld-¹ (X(C)⁹C) A l b ( X ) ( C ) -+ —2JZT(

H^za~^l(X(C)⁹Z(d)) + F^a

such that the complex Abel-Jacobi map cl' on CH^d(X)⁰ can be identified w i t h a^x . Hence M u m f o r d ' s theorem implies the following: I f X is a com- plex surface w i t h Z Z²( X , ff^x) / O, then the Abel-Jacobi map

cl': CH²(X)⁰ % A l b ( X ) ( C ) - H³(X(C)⁹C)

0 Z Z³( X ( C ) , Z(2)) + F²

has a huge kernel. (In fact, by R o i t m a n ' s isomorphism (1.8), K e r c / ' is torsion-free and divisble, and hence, a Q-vector space. I f it had a finite basis, it w o u l d lie i n the image o f g^: P i c⁰( C^x) C Z Z²( X )⁰ for some morphism g: C^f —> X o f a smooth projective curve C' into X , and we w o u l d obtain

1.6(ii), a contradiction). Moreover, C Z Z²( X )⁰ cannot be given the structure

of (the group o f points of) an abelian variety, i n a reasonable way. In fact, correctly interpreted this w o u l d mean that a^x is an isomorphism (since a^x is universal for regular maps (cf. [BS, p. 1238]) into abelian varieties).

B l o c h proposed the following converse o f M u m f o r d ' s theorem.

CONJECTURE

1.8 [Bll, Lecture 1]. If H²(X,@^X) = 0, then a^x: CH²(X)⁰

—• A l b ( X ) ( C ) is an isomorphism.

Moreover, he extended both M u m f o r d ' s theorem and this conjecture to arbitrary base fields, by using ^-adic cohomology. N a m e l y , he observed the following equivalence for the surface X/C :

/ Z²( X⁵^^r) = O,

<* H²{X(C), C) = H¹ ' ¹ [by Hodge theory],

H ( X ( C ) , C ) is algebraic, i.e., generated by cycle classes o f divisors [by Lefschetz's theorem].

The last statement makes sense for any W e i l cohomology, and B l o c h proves Theorem 1.9 and makes Conjecture 1.10 [Bll, Lecture 1].

THEOREM

^{1.9. Let X} be a smooth projective surface over a field k, and let Q D k be a universal domain. If a^x: CH (X^Q)⁰ —• A l b ( X ) ( Q ) is an

2

isomorphism, then H ( X , Q^(I)) is algebraic for £ ^ char(/c).

CONJECTURE

1.10. The converse holds', i.e., if H²(X, Q^E(I)) is algebraic

2

for some £ / char(/c), then CH (X) is representable.

REMARKS

1.11. (a) The consideration o f a universal d o m a i n or at least o f a "field containing many parameters" is essential here. In fact, the homo- morphism a y : CH^dimw(X)⁰ -» A\b(X)(k) is k n o w n to be an isomorphism for finite fields [ K S , §9] and is conjectured to be an isomorphism for number fields k by B l o c h and Beilinson (cf. [Be3, 5.2]).

(b) I f k = C = Q , then Conjectures 1.8 and 1.10 are equivalent, by the canonical comparison isomorphism between singular and etale cohomology.

(c) We have seen that K e r a ^ = K e r c / ' for the complex Abel-Jacobi map cl' on CH^D(X)⁰ . I f k is finitely generated and £ ^ char(/c), then one can show as well (cf. [Ja3, 9.14]) that (Kera^x) <8> Z^I = ( K e r c / ' ) <g> Z^I for the

£-adic Abel-Jacobi map cl' on CH^D(X)⁰.

Conjecture 1.10 is k n o w n to be true for an abelian surface X over a field k

2

of characteristic p > 0 . In fact, since Tate's conjecture for H ( X , Q^(I)) is known by work o f Z a r h i n [Za] and M o r i [ M o ] , one easily sees that H²(X,Q^E(I)) is algebraic i f and only i f X is isogeneous to a product o f two supersingular elliptic curves. The representability o f CH (X)⁰ for such surfaces was proved by M a r u y a m a and Suwa [ M S , Theorem 3.2].

In general, B l o c h proposes the following strategy. H e introduces the fol- lowing three-step filtration

CH²(X) U |

(1.9) CH²(X)⁰ = K e r ( d e g : CH²(X) -> Z) U |

T(X) = K e r ^ : CH²(X)⁰ -» Alb(X)(Ic))

The action o f correspondences, i.e., o f CH (X x X), respects this filtration, and one can show

THEOREM

1.12 (cf. [Bll, 1.11]). Iftheactionofcorrespondenceson T(XQ) factors through homological equivalence, then Conjecture 1.10 is true.

F i n a l l y , B l o c h discusses various aspects o f the mysterious relation T(X)

~

^H2.

In the next section we review how B e i l i n s o n "explains" this relation o n the basis o f m i x e d motives. H i s approach heads toward an understanding o f arbitrary C h o w groups (in fact, arbitrary m o t i v i c cohomology), even i n the nonrepresentable case.

2. Beilinson⁹S formula

In his paper o n height pairings [Be3, 5.10], Beilinson stated a conjecture on m i x e d m o t i v i c sheaves that leads to the following explicit conjecture i n w h i c h no m i x e d motives are mentioned.

CONJECTURE

2.1 (Version 1 o f Beilinson's conjecture). Let k be a field.

For every smooth projective variety X over k there exists a descending filtra- tion F on CH³(X)^Q, for all j > 0, such that

(a) F0CHJ(X)Q = CHJ(X)Q, FxCHJ(X)Q = CHj(X)HOM q , for some fixed Weil cohomology H'(X);

(b) FrCHI(X)Q . FsCHJ(X)Q C Fr+sCHI+J(X)Q under the intersection product;

(c) F' is respected by f* and f^ for morphisms f: X -» Y \

(d) (assuming the algebraicity of the Kunneth components of the diagonal) GX^VFCH^J(X)^Q depends only on the motive modulo homological equivalence h2J~V(X)\ and

(e) FuCHJ(X)Q = Q for i / » 0 .

The meaning o f (d) is as follows. B y (b) and (c) the action o f correspon- dences respects the filtration F', a n d by (a) (applied to X x X) the induced action on GT^vfCH^j(X)^Q = F^vCH^J(X)^qJF^v+^xCH^j'(X)^Q factors through ho- mological equivalence (i.e., CH^d(X x X)^HOM acts as zero). F o r defining the

motive H¹(X) we have to assume that the K i i n n e t h components Ttⁱ eH^2d~^l(X)® Hⁱ(X), d = dimX,

o f the diagonal A are algebraic ([K13, conjecture C(X)]; this is a fairly weak consequence o f the standard conjectures or the Tate conjecture (cf. [Ta, §3]) and holds for varieties over finite fields [KM, T h e o r e m 2]). T h i s means that the idempotent

H (X)-> Hⁱ(X) -> H (X)

is given by an algebraic correspondence (again denoted by Tiⁱ) w h i c h is unique and an idempotent i n CH^d(X x X )^Q/ ~^hom . T h e motive h^l(X) can then be defined as the triple (X, Tiⁱ, 0) (cf. [Scho]), a n d (d) means that (2.1) TtⁱIGr^ufCH^j (X)^q = S^ilj^- i d ,

where S^{a b} is the K r o n e c k e r symbol.

Let us write this out i n more detail, for further reference. R e c a l l [Scho]

that the category Jt^ o f motives over k m o d u l o some adequate equiva- lence relation ~ can be defined as follows. Objects are triples (X, p, m) = ( X , p, m)^ , where X is smooth projective over k, p¹ = p e C o r r ° ( X , X) is an idempotent, and m e Z , and we have

H o m ( ( X , p, m),{Y,q⁹ n)) = qCorrⁿ-^m(X, Y)p.

Here C o r r^r( X , Y) = QⁱCH^d^^r(Xⁱ x y )^Q/ ~ , for X = UXⁱ > ^xt ^{o f} P^{u r e} dimension d^{, is the group o f correspondences o f degree r, and c o m p o s i t i o n is the one o f correspondences recalled i n § 1 . I f h(X) = (X, i d , 0 )^{h o m} is the motive m o d u l o homological equivalence associated to X, then

2d

H(X) = Qhⁱ(X),

i=0

since the Tiⁱ are pairwise orthogonal idempotents i n CH^d(X x X )^Q/ ~^{h o m}= EndZz(X) and

2d 2d

(2.2) EndZz(X) = 0 E n d Z z^z' ( X ) = QnfH^d (X x X y ~ ^{h o m} Ti^l,

i=0 /=O

since the Tiⁱ are central. N o w (2.1) and (d) both mean that the action o f EndZz(x) o n Gr^vpCH^j(X)^q factors through the quotient E n d Z z ' ( X ) .

One w o u l d i n fact expect the following stronger f o r m o f 2.1 to be true:

STRONG CONJECTURE

2.1. This is the same as Conjecture 2 . 1 , except that (e) is replaced by

(strong e) F^j+X CH^j(X)^q = 0 .

LEMMA

2.2. If the standard conjecture of Lefschetz type B(X) is true, then Conjecture 2.1 implies its strong form.

PROOF

(Compare [Ja3, 11.2]). B y B(X) there is a hard Lefschetz isomor- p h i s m o f motives

L^d"⁵: h^s(X) ^ h^ld~^s(X)(d - s) (s<d),

where M(n) is the n-fold Tate twist o f a motive (in the notation o f [Scho] we have M(n) = M® L®~ⁿ and h^2d~^s(X)(d - s) = (X, n^2d_^s, d - s)). Hence n^s: h(X) —• h^s(X) —• h(X) factors as a c o m p o s i t i o n n^s = a o a, where a e n^2d_^sCH^ld-^s(X x X )^Q/ ~^{h o m}= Hom(h(X), h^2d~^s(X)(d - s)) and cl G CH^s(XxX)^Q/^^hom= Hom(h^2d~^s(X)(d-s), h(X)). I f a is represented by an element T e CH^2d~^s(X x X)^q , then T maps CH^j(X)^q to CH^d~^s+j(X)^q :

C Z Z⁷( X )^{Q P}4 C t f⁷ ( X X X)^q ± CH^2d~^s+J(X x X)^{q ( P} ^ CH^d^^j(X)^q, and this group vanishes for 5 < j . B y (2.1), we conclude that G r J l C / /⁷( X )^Q

= 0 for 1/ > j , hence the result. W e note that L e m m a 2.2 makes sense and stays true, i f we restrict the consideration o f Conjecture 2.1 to X and X x X .

A relation w i t h m i x e d motives is given by the following version o f B e i l i n - son's conjecture. F r o m now on, let Jf^k = Jf^kom be the category o f mo- tives over k m o d u l o homological equivalence. We sometimes call these Grothendieck motives, since they are the object o f his standard conjectures, but, o f course, Grothendieck also defined and considered other variants.

CONJECTURE

2.3 (Version 2 o f Beilinson's conjecture). This is the same as Conjecture 2 . 1 , except that (d) is replaced by

(d') (assumptions as in Conjecture 2.1(d)) There is an abelian category JifJf^k (of "mixed motives over k ") containing the category Jf^k of Grothen- dieck motives as a full subcategory, and a functorial isomorphism

(2.3) Gv^ufCH^j(X)^q S E x t ^ ( 1 , H^2j^(X)(J)), where 1 = A(SpecZc) is the trivial motive.

In the following we call (2.3) "Beilinson's formula". It makes more evi- dent and precise (granting the existence o f JfJf^k !) how Gr^u7CH^j(X)^q de- pends o n the motive h^2j~^u(X). N o t e that the category Jf^k is expected to be semisimple, so there are no n o n t r i v i a l (Yoneda) extension groups E x t ^ ( - , - ) for / > 1. T h e existence o f n o n t r i v i a l extensions is a specific feature o f the

" m i x e d " situation.

2.4. Let us consider Beilinson's formula for v = 0 . Since 1 and h^2j(X)(j) are i n Jf^k, the assumption that Jf^k is a full subcategory o f JfJf^k means that we have an i s o m o r p h i s m

H o m ^( 1 , Zi^{2 7}(X)O')) - H o m ^( 1 , h^2j(X)(J)).

O n the other hand, by (a) we want

GT^0FCH^J(X)^Q = C H^J (X)^Q/~^HOM. But the equality

CH^J(X)^Q^^HOM= H o m ^ l , h^2j (X)(J))

holds by construction o f the category Jf^k ; by definition, the left-hand side is H o r n ^ ( 1 , h(X)(j)), a n d this is equal to H o r n ^ ( 1 , h^2j(X)(j)), since TiⁱCH^j(X)^q^^homC TiⁱH^2j(X) = 0 for i ±2j.

For u = l we want a m a p

(2.4) F¹CH^J(X)^Q = CH^J(X)^HOM^ - E x t ^ ( 1 , h^2J-\X)(j)) whose kernel is F CH^j(X)^q. W h i l e a c a t e g o r y JfJf^k as i n (d') has not yet been identified (but cf. [Li] for a discussion o f several proposals), we have already encountered the reflection o f (2.4) i n cohomology. F o r example, con- sider the ^-adic cohomology, t ^ char(/c). B y the well-known isomorphisms

(2.5) H^v(G^kiV)^ExX^10k(QⁱⁱV)

for any (finite-dimensional) -representation V o f G^k, we can i n fact reinterprete the I-adic Abel-Jacobi map (after tensoring with Q a n d Qⁱ) as a m o r p h i s m

(2.6) cf: CH^j(X)^homq -> E x t ^ ( Q , , H^2j'¹ (X, QⁱW)),

the I -adic version o f (2.4). I f k = C⁹ then we obtain something s i m i l a r i n the Hodge theory; particularly, i f M H S is the category o f m i x e d Hodge structures [De2, 2.3; St], then C a r l s o n [Ca, P r o p o s i t i o n 2] and Beilinson [Be2, 1.7] have constructed an isomorphism

(2.7) E x 4^{H S}( Z , H) = W⁰H^C/(W⁰H + F⁰W⁰H^C)

for a m i x e d Hodge structure H, where W⁹ is the ascending weight filtration.

The same formula holds for E x t ^ _^{M H S}( ^ , H), where A is Q or R , ^ - M H S is the category o f ^4-Hodge structures, a n d H is i n y l - M H S (loc. cit.). Since H^2j~^{(X(C)⁵QO')) is o f weight - 1 < 0 , we can thus regard the complex Abel-Jacobi map (after tensoring w i t h Q) as a m o r p h i s m

(2.8) cf: CH^J(X)^HOMQ - E x t^ .^{M H S}( Q , H^2J~^L (X(C), Q(J))).

T h e m a p (2.4) should be compatible w i t h the Abel-Jacobi maps into co- homology; i.e., these should factor as

H¹ (G^K , H^2J-¹ (X^FQ^tU))) = ExtJ, ( Q , , H^2J-\X, Q^e(J)))

CH^J(X)^HOMQ ^ E x t ^ d , h^2J-\X)(j))

\ i H^B

H^2J-¹(X(C),C) _ , H^2J-^L(XO(i))) H^2J-^X(ClQU))+ F^J ~ ^{E X T} ^ ^{S I Q} ' ^{H { X M J ) ) )} -

H e r e the vertical maps should be induced by exact, faithful " r e a l i z a t i o n "

functors

H^£: JfJf^k —• Rep(G^k, Qⁱ) := category o f finite-dimensional

(2.10) Q^-representations o f G^k ,

H^B: JfJf^k ^ Q - M H S ,

w h i c h extend the existing functors Hⁱ and H^B on Jf^k induced by I -adic a n d singular cohomology, respectively. N o t e that by definition we have

Hⁱ(I) = Qⁱ, H^I(Hⁱ(X)U)) = TiⁱHCX⁹ Qⁱ(J)) = Hⁱ(X⁹ Qⁱ(J)), H^B(I) = Q⁹ H^b(H¹(X)U)) = H¹(X(C)⁹Q(J))⁹

where Q^ a n d Q are the t r i v i a l Q^-representation and t r i v i a l Q-Hodge struc- ture, respectively. N o t e also that the corresponding diagram for v = 0 , (2.11) _

H^2j(X⁹ Qⁱ(J))⁰* = H o m^{c 7 4}( Q , , H^2j(X⁹ Qⁱ(J)))

/ U^e

CH^j(X)^q - CH^j(X)^q^^hom = H o m^ l , h^2j(X)(J))

\ I n^B

H^2j(X(C)⁹Q(J))DH^jj = Hom^^m(Q⁹H^2j(X(C)⁹Q(J)))⁹ exists and commutes by definition.

2.5. T h e extensions given by Abel-Jacobi maps can i n fact be constructed i n a universal and geometric way: I f z G Z^j(X) is a cycle that is homolog- ically equivalent to zero, let Z be the support o f z and put U = X - Z . F o r the I -adic cohomology (£ ^ char k) we obtain a commutative diagram i n the category o f Q^-representations o f G^k

O _» H²^ⁱ(X⁹QⁱU)) - H^2j-^lW⁹Qⁱ(J)) - Hg(X⁹QⁱU)) -H^{l j}( X⁹Q^tU ) ) U| | z cl{z)=0

O -> i f² ' -¹^ ^ ) ) - - - O

Here the exact top row is part o f the localization sequence [Mi, III, 1.25], where by purity H^~^l(X⁹ Qⁱ(J)) = O and H±^J(X, Qⁱ(J)) * Qⁱ as a Q ^r representation o f G^k , where A is the set o f irreducible components o f Z . T h i s gives the map denoted by z (mapping 1 e Qⁱ to the local cycle class o f z ; cf. [Mi, V I , §9]), a n d the b o t t o m exact sequence is obtained by pull-back v i a z . It is shown i n [Ja3, 9.4] that the extension class o f this sequence is the image o f z under the map cl' i n (2.6).

A n analogous result holds for the complex version (2.8), by using the cor- responding diagram o f m i x e d Q-Hodge structures given by singular c o h o m o l - ogy (loc. cit., 9.2 and 9.7c)). T h i s suggests that there should exist a similar diagram i n JfJf^k

O - h^2j-\X)(j) -> h^2j-\u)(j) - H^2j(X)(J) -> H^lj(X)(J) Il U | T

O - h^2j-\X)(J) -> E I -+ O