As we mentioned above, the Chow groups are an analogue for the even cohomology
of a smooth projective variety over *C*. To make this relation more precise
we first examine the cycle class map
(*X*)(*X*,*Z*).

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

A similar assertion for
(*X*)(*X*,*Z*) when *n* = dim *X* is
obvious since this homomorphism is surjective. Thus one could conjecture that
the image of
(*X*)(*X*,*Z*) consists precisely of the subgroup
of Hodge classes

(*X*,*Z*) = ker((*X*,*Z*)(*X*,*C*)/*F*^{p}(*X*,*C*))

However, this is known to be false unless we tensor with
(*X*) *Q*(*X*,*Q*) is surjective

is the celebrated
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
(*X*,*C*) has no end terms.

Beilinson has generalised this as follows. We define the level filtration
*L*^{p}(*X*,*Q*) as the intersection the kernels of the restriction
homomorphisms
(*X*)(*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}(*X*,*Q*) (*X*,*Q*).
Then (conjecturally) the complexity of
(*X*) *Q* is
``measured'' by *k*(*p*). In particular, he conjectures that (a) if
*k*(*p*) = 0, then
(*X*) *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
*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 *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}(*X*,*Q*)*L*^{p}(*Y*,*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
*CH*^{1}(*X*)*CH*^{1}(*Y*) is an isomorphism for
*n* = dim *Y* 3, and an
inclusion for *n* = 2.

A more refined analysis using the monodromy of Lefschetz pencils shows
that for *n* = 2, we have
*CH*^{1}(*X*) *CH*^{1}(*Y*) provided *Y* is a *general* hyperplane section; this is called the *Noether-Lefschetz
theorem*. Similarly, it is known that for *n* = 1,
*CH*^{1}(*X*)*CH*^{1}(*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].