7 Relation with cohomology

As we mentioned above, the Chow groups are an analogue for the even cohomology of a smooth projective variety over $\Bbb$C. To make this relation more precise we first examine the cycle class map $\CH^{p}_{}$(X)$\to$$\HH^{2p}_{}$(X,$\Bbb$Z).

The theorem of Lefschetz on (1, 1) classes asserts that the image of the cycle class map $\CH^{1}_{}$(X)$\to$$\HH^{2}_{}$(X,$\Bbb$Z) is precisely the kernel of $\HH^{2}_{}$(X,$\Bbb$Z)$\to$$\HH^{2}_{}$(X,$\Bbb$C)/F¹$\HH^{2}_{}$(X,$\Bbb$C), where F^* denotes the Hodge filtration. Equivalently, with respect to the Hodge decomposition $\HH^{m}_{}$(X,$\Bbb$C) = $\oplus_{p+q=m}^{}$ $\HH^{p,q}_{}$(X) into spaces $\HH^{p,q}_{}$ of harmonic forms of type (p, q), the classes of divisors in $\HH^{2}_{}$(X,$\Bbb$Z) are precisely those classes $\alpha$ such that $\alpha_{{\Bbb C}}^{}$ $\in$ $\HH^{2}_{}$(X,$\Bbb$C) lies in $\HH^{1,1}_{}$(X).

A similar assertion for $\CH^{n}_{}$(X)$\to$$\HH^{2n}_{}$(X,$\Bbb$Z) when n = dim X is obvious since this homomorphism is surjective. Thus one could conjecture that the image of $\CH^{p}_{}$(X)$\to$$\HH^{2p}_{}$(X,$\Bbb$Z) consists precisely of the subgroup of Hodge classes

$$\Hg^{p}_{}$$(X,$$\Bbb$$Z) = ker($$\HH^{2p}_{}$$(X,$$\Bbb$$Z)$$\to$$$$\HH^{2p}_{}$$(X,$$\Bbb$$C)/F^p$$\HH^{2p}_{}$$(X,$$\Bbb$$C))

However, this is known to be false unless we tensor with $\Bbb$Q. Early counter-examples are in [2], and more recently one knows from the work of Kollár, Mori, Miyaoka [22] that a general hypersurface of degree 125 in $\Bbb$P⁴ cannot contain a cycle of degree 1. The assertion:

$$\CH^{p}_{}$$(X) $$\otimes$$ $$\Bbb$$Q$$\to$$$$\Hg^{p}_{}$$(X,$$\Bbb$$Q)  is surjective

is the celebrated Hodge conjecture. Many special cases are known but there is no general theorem (or even heuristic) in this direction.

The second relation of Chow groups to cohomology is via Griffiths' Abel-Jacobi homomorphism to the points of the Intermediate Jacobian. As we saw with zero cycles, the kernel of this homomorphism can be very large. On the other hand Bloch's conjecture asserts (in the case of zero cycles) that the kernel is torsion (hence zero by Roitman's theorem) in case the Hodge decomposition of $\HH^{2n-k}_{}$(X,$\Bbb$C) has no end terms.

Beilinson has generalised this as follows. We define the level filtration L^p$\HH^{2p-k}_{}$(X,$\Bbb$Q) as the intersection the kernels of the restriction homomorphisms $\HH^{2p-k}_{}$(X)$\to$$\HH^{2p-k}_{}$(Y), where Y runs over all subvarieties of X of dimension less than p. Let k(p) be the largest integer such that L^p$\HH^{2p-k}_{}$(X,$\Bbb$Q) $\neq$ $\HH^{2p-k}_{}$(X,$\Bbb$Q). Then (conjecturally) the complexity of $\CH^{p}_{}$(X) $\otimes$ $\Bbb$Q is ``measured'' by k(p). In particular, he conjectures that (a) if k(p) = 0, then $\CH^{p}_{hom}$(X) $\otimes$ $\Bbb$Q = 0, and (b) if k(p) = 1, then the kernel of the Abel-Jacobi homomorphism should be torsion.

For X a complete intersection subvariety of $\Bbb$P^N we have the result of Esnault-Nori-Srinivas [13] which generalises earlier work of Deligne and Deligne-Dimca, computing the width of the Hodge structures. From these results one can show that for a fixed p and multidegree, and for sufficiently large N (made precise by their results), we have k(p) = 0 (if Grothendieck's generalised Hodge conjecture is true). The Chow groups of these varieties ought to be $\Bbb$Z upto torsion. One can see that Roitman's theorem on zero cycles precisely achieves the predicted result. For higher dimensional cycles a weak form of this conjecture is known; see [24], [31].

Another way of examining the consequences of this conjecture is to look at the situation of a smooth subvariety Y of X such that the restriction map on cohomology induces an isomorphism L^p$\HH^{2p-k}_{}$(X,$\Bbb$Q)$\to$L^p$\HH^{2p-k}_{}$(Y,$\Bbb$Q). One such case is again the complete intersection situation, where the Leftshetz hyperplane section theorems give us such isomorphisms. In this case the Grothendieck-Lefschetz theorem precisely achieves the bound predicted. This theorem asserts that the restriction map CH¹(X)$\to$CH¹(Y) is an isomorphism for n = dim Y $\geq$ 3, and an inclusion for n = 2.

A more refined analysis using the monodromy of Lefschetz pencils shows that for n = 2, we have CH¹(X) $\cong$ CH¹(Y) provided Y is a general hyperplane section; this is called the Noether-Lefschetz theorem. Similarly, it is known that for n = 1, CH¹(X)$\to$CH¹(Y) is injective for general Y.

The higher dimensional analogue of the Grothendieck-Lefschetz theorem was posed as a problem by Hartshorne [19] (well before Beilinson formulated his conjectures). Only very weak results are known in this direction [31]. Higher dimensional analogues of the Noether-Lefschetz theorems have recently been formulated by Nori [29] with cohomological justification rather similar to the Beilinson conjectures. Some very weak statements along these lines are known [15], [40], [21].