Introduction

These lectures are devoted to the interplay between cohomology and Chow groups, and also to the consequences, for the topology of a family of smooth projective varieties, of statements concerning Chow groups of the general or very general fiber.

A crucial notion is that of the coniveau of a cohomology. A Betti cohomology class has geometric coniveau = c if it is supported on a closed algebraic subset of codimension = c. The coniveau of a class of degree k is = k/2. As a smooth projective variety X has nonzero cohomology in degrees 0, 2, 4, ... obtained by taking c1(L), with L an ample line bundle on X, and its powers c1(L)i, it is not expected that the whole cohomology of X has large coniveau. But it is quite possible that the "transcendental" cohomology H*B(X)[perpendicular to]alg, consisting of classes orthogonal (with respect to the Poincaré pairing) to cycle classes on X, has large coniveau.

There is another notion of coniveau: the Hodge coniveau, which is computed by looking at the shape of the Hodge structures on H*B (X;Q). Classes of algebraic cycles are conjecturally detected by Hodge theory as Hodge classes, which are the degree 2k rational cohomology classes of Hodge coniveau k. The generalized Hodge conjecture due to Grothendieck more generally identifies the coniveau above (or geometric coniveau) to the Hodge coniveau.

The next crucial idea goes back to Mumford, who observed that for a smooth projective surface S, there is a strong correlation between the structure of the group CH0(S) of 0-cycles on S modulo rational equivalence and the spaces of holomorphic forms on S. The degree 1 holomorphic forms govern the Albanese map, which itself provides us with a certain natural quotient of the group CH0(S)hom of 0-cycles homologous to 0 (that is, of degree 0 if S is connected), which is in fact an abelian variety. This part of CH0(S)hom is small in different (but equivalent) senses, first of all because it is parametrized by an algebraic group, and second because, for any ample curve C [subset] S, the composite map

CH0(C)hom -> CH0(S)hom -> Alb(S)

is surjective. Thus 0-cycles supported on a given ample curve are sufficient to exhaust this part of CH0(S)hom.

Mumford's theorem says the following.

Theorem 1.1 (Mumford 1968). If H2;0(S) ? 0, no curve C [??] S satisfies the property that j*: CH0(C) -> CH0(S) is surjective.

The parallel with geometric coniveau in cohomology is obvious in this case; indeed, the assumption that H2;0(S) ? 0 is equivalent (by the Lefschetz theorem on (1; 1)-classes) to the fact that the cohomology H2B (S, Q) is not supported on a divisor of S. Thus Mumford's theorem exactly says that if the degree 2 cohomology of S is not supported on any divisor, then its Chow group CH0(S) is not supported on any divisor.

The converse to such a statement is the famous Bloch conjecture. The Bloch conjecture has been generalized in various forms, one involving filtrations on Chow groups, the graded pieces of the filtration being governed by the coniveau of Hodge structures of adequate degree (see [58], [89], and Section 2.1.4)). The crucial properties of this conjectural filtration are functoriality under correspondences, finiteness, and the fact that correspondences homologous to 0 shift the filtration.

We will focus in these notes on a more specific higher-dimensional generalization of the Bloch conjecture, "the generalized Bloch conjecture," which says that if the cohomology H*B (X, Q)[perpendicular to] alg has coniveau = c, then the cycle class map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is injective for i = c - 1. In fact, if the variety X has dimension > 2, there are two versions of this conjecture, according to whether we consider the geometric or the Hodge coniveau. Of course, the two versions are equivalent assuming the generalized Hodge conjecture. In Section 4.3 we will prove this conjecture, following, for the geometric coniveau and for very general complete intersections of ample hypersurfaces in a smooth projective variety X with "trivial" Chow groups, that is, having tttthe property that the cycle class map

cl : CH*(X)Q -> H2*B (X, Q)

is injective (hence an isomorphism according to).

A completely different approach to such statements was initiated by Kimura, and it works concretely for those varieties that are dominated by products of curves. It should be mentioned here that all we have said before works as well in the case of motives (see Section 2.1.3). In the above-mentioned work of Kimura, one can replace "varieties that are dominated by products of curves" by "motives that are a direct summand of the motive of a product of curves." In our paper, we can work with a variety X endowed with the action of a finite group G and consider the submotives of G-invariant complete intersections obtained by considering the projectors Gp [member of] CH(Y × Y)Q associated via the action of G on Y to projectors p [member of] Q[G].

An important tool introduced by Bloch and Srinivas in [15] is the so-called decomposition of the diagonal. It relates information concerning Chow groups CHi(X), for small i, to the geometric coniveau of X. Bloch and Srinivas initially considered the decomposition of the diagonal in its simplest form, starting from information on CH0(X), and this has subsequently been generalized in to a generalized decomposition of the diagonal. This leads to an elegant proof of the generalized Mumford–Roitman theorem, stating that if the cycle class map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is injective for i = c - 1, then the transcendental cohomology H*B (X, Q)[perpendicular to] alg has geometric coniveau = c. (The generalized Bloch conjecture is thus the converse to this statement.)

The study of the diagonal will play a crucial role in our proof of the generalized Bloch conjecture for very general complete intersections. The diagonal will appear in a rather different context in Chapter 5, where we will describe our joint work with Beauville and further developments concerning the Chow rings of K3 surfaces and hyper-Kähler manifolds. Here we will be concerned not with the diagonal ?X [subset] X × X but with the small diagonal ? [congruent to] X [subset] X × X × X. The reason is that if we consider ? as a correspondence from X × X to X, we immediately see that it governs, among other things, the ring structure of CH*(*). In we obtained for K3 surfaces X a decomposition of ? involving the large diagonals, and a certain canonical 0-cycle o canonically attached to X.

We will show in Section 5.3 an unexpected consequence, obtained in, of this study combined with the basic spreading principle described in Section 3.1, concerning the topology of families of K3 surfaces.

In a rather different direction, in the final chapter we present recent results concerning Chow groups and Hodge classes with integral coefficients. Playing on the defect of the Hodge conjecture for integral Hodge classes (see [5]), we exhibit a number of birational invariants which vanish for rational projective varieties and are of torsion for unirational varieties. Among them is precisely the failure of the Bloch–Srinivas diagonal decomposition with integral coefficients: in general, under the assumption that CH0(X) is small, only a multiple of the diagonal of X can be decomposed as a cycle in X × X. The minimal such multiple appears to be an interesting birational invariant of X.

In the rest of this introduction, we survey the main ideas and results presented in this monograph a little more precisely. Background material is to be found in Chapter 2.

1.1 DECOMPOSITION OF THE DIAGONAL AND SPREAD

1.1.1 Spread

The notion of the spread of a cycle is very important in the geometric study of algebraic cycles. The first place where it appears explicitly is Nori's paper, where it is shown that the cohomology class of the spread cycle governs many invariants of the cycle restricted to general fibers. The idea is the following (see also): Assume that we have a family of smooth algebraic varieties, that is, a smooth surjective morphism

p : ? -> B,

with geometric generic fiber ?[bar.?] and closed fiber ?s. If we have a cycle Z [member of] Zk(?[bar.?]), then we can find a finite cover [??] -> U of a Zariski open set U of B such that Z is the restriction to the geometric generic fiber of a cycle [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If we are over C, we can speak of the spread of a cycle Zs [member of] Zk(Xs), where s [member of] B is a very general point. Indeed, we may assume that p is projective. We know that there are countably many relative Hilbert schemes Mi -> B parametrizing all subschemes in fibers of p. Cycles Z = [summation]i niZi in the fibers of p are similarly parametrized by countably many varieties pJ : NJ -> B, where the pJ's are proper, and the indices J also encode the multiplicities ni.

Let B' [subset] B be the complement of the union [union]J [member of] E Im pJ, where E is the set of indices J for which pJ is not surjective. A point of B' is a very general point of B, and by construction of B', for any s [member of] B', and any cycle Zs [member of] Zk(Xs), there exist an index J such that MJ ! B is surjective, and a point s0 2 MJ such that pJ (s') = s, and the fiber [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at s' of the universal cycle

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

parametrized by MJ, is the cycle Zs. By taking linear sections, we can then find M'J [subset] MJ, with s' [member of] M'J, such that the morphism M'J -> B is dominating and generically finite. The restriction [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the universal cycle [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to ? × B M'J is then a spread of Zs.

1.1.2 Spreading out rational equivalence

Let p : ? -> B be a smooth projective morphism, where B is smooth irreducible and quasi-projective, and let Z [subset] X be a codimension k cycle. Let us denote by Zt [subset] Xt the restriction of Z to the fiber Xt. We refer to Chapter 2 for the basic notions concerning rational equivalence, Chow groups, and cycle classes.

An elementary but fundamental fact is the following result, proved in Section 3.1.

Theorem 1.2 (See Theorem 3.1). If for any t [member of] B the cycle Zt is rationally equivalent to 0, there exist a Zariski open set U [subset] B and a nonzero integer N such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is rationally equivalent to 0, where ?U := p-1(U).

Note that the set of points t [member of] B such that Zt is rationally equivalent to 0 is a countable union of closed algebraic subsets of B, so that we could in the above statement, by a Baire category argument, make the a priori weaker (but in fact equivalent) assumption that Zt is rationally equivalent to 0 for a very general point of B.

This statement is what we call the spreading-out phenomenon for rational equivalence. This phenomenon does not occur for weaker equivalence relations such as algebraic equivalence.

An immediate but quite important corollary is the following.

Corollary 1.3. In the situation of Theorem 1.2, there exists a dense Zariski open set U [subset] B such that the Betti cycle class [Z] [member of] H2kB (?, Q) vanishes on the open set ?U.

The general principle above applied to the case where the family ? -> B is trivial, that is, ? [congruent to] X × B, leads to the so-called decomposition principle due to Bloch and Srinivas. In this case, the cycle Z [subset] B × X can be seen as a family of cycles on X parametrized by B or as a correspondence between B and X. Then Theorem 1.2 says that if a correspondence Z [subset] B × X induces the trivial map

CH0(B) -> CHk(X); b -> Zb,

then the cycle Z vanishes up to torsion on some open set of the form U × X, where U is a dense Zariski open set of B.

The first instance of the diagonal decomposition principle appears in [15]. This is the case where X = Y \ W, with Y smooth and projective, and W [subset] Y is a closed algebraic subset, B = Y, and Z is the restriction to Y × (Y \ W) of the diagonal of Y. In this case, to say that the map

CH0(B) -> CH0(X); b -> Zb,

is trivial is equivalent to saying, by the localization exact sequence (2.2), that any point of Y is rationally equivalent to a 0-cycle supported on W. The conclusion is then the fact that the restriction of the diagonal cycle ? to a Zariski open set U × (Y \ W) of Y × Y is of torsion, for some dense Zariski open set U [subset] Y. Using the localization exact sequence, one concludes that a multiple of the diagonal is rationally equivalent in Y × Y to the sum of a cycle supported on Y × W and a cycle supported on D × Y, where D := Y \ U. Passing to cohomology, we get the following consequence.

Corollary 1.4. If Y is smooth projective of dimension n and CH0(Y) is supported on W [subset] Y, the class [?Y] [member of] H2nB (Y × Y, Q) decomposes as

[?Y] = [Z1] + [Z1];

where the cycles Zi are cycles with Q-coefficients on Y × Y , Z1 is supported on D × Y for some proper closed algebraic subset D [??] Y, and Z2 is supported on Y × W.

1.1.3 Applications of Mumford-type theorems

In the paper by Bloch and Srinivas, an elegant proof of Mumford's theorem (Theorem 1.1) is provided, together with the following important generalization.

Theorem 1.5 (Roitman 1980, Bloch and Srinivas 1983; see Theorem 3.13). Let X be a smooth projective variety and W [subset] X be a closed algebraic subset of dimension = k such that any point of X is rationally equivalent to a 0-cycle supported on W. Then H0(X, OlX) = 0 for l > k.

This theorem, together with other very important precisions concerning the coniveau (see Section 2.2.5) of the cohomology of X, is obtained using only the cohomological decomposition of the diagonal of X, that is, Corollary 1.4.