Motives and filtrations on Chow groups
Let \(\text{CH}^r(X)\) be the Chow group of algebraic cycles of codimension \(r\) on a smooth complex projective algebraic variety \(X\). According to a conjecture of A. Beilinson and S. Bloch, there should exist a filtration \(\text{CH}^r(X) = F_{\mathcal M}^0\supset \ldots F_{\mathcal M}^\nu\supset\ldots\) which satisfies the following conditions: (1) It is respected by the action of every algebraic correspondence \(\Gamma\) on \(X\); (2) the induced action of \(\Gamma\) on each graded piece depends only on a certain Künneth component of the cohomology class of \(\Gamma\); (3) there exists an integer \(N > 0\) (depending on \(X\)) such that \(F_{\mathcal M}^\nu = 0\). The main result of this paper gives a construction of a certain filtration \((F_H^\nu)\) on \(\text{CH}^r(X)\) which satisfies properties (1) and (2) from above. The conjectural theory of motives, if true, shows that the filtration constructed in the paper coincide with the filtration \((F_{\mathcal M}^\nu)\). The filtration \((F_H^\nu)\) sheds some light on the finiteness properties of the Chow groups. The author introduces a certain rank \(r\) of a family of algebraic cycles viewed as a measure of non-representability of the family. For example, \(\text{Ker}(\rho^i)\otimes{\mathbb{Q}} = 0\) implies that \(r(A^i(X))\leq 1\). Here \(A^i(X)\) is the subgroup of \(\text{CH}^i(X)\) formed by cycle classes which are algebraically equivalent to zero and \(\rho^i:A^i(X)\to \text{Jac}^i(X)\) is the Abel-Jacobi map. As a generalization of most of the previous work on non-representability of Chow groups the author proves that, for any algebraic correspondence \(\Gamma\), \(\text{Im(Gr}_{F_H}^\nu(\Gamma_*)) = 0\) implies that \(r(\text{Im Gr}_{F_H}^\nu(\Gamma_*))\leq \nu-1\). The two conditions are equivalent if \(\nu\leq 2\) or one of the standard Grothendieck conjectures holds.
scientific article; zbMATH DE number 885524