Regular homomorphisms and mixed motives

Abstract

Let $X$ be a smooth projective variety of dimension $d$ over an algebraically closed field $k$. The main goal of this paper is to study, in the context of Voevodsky's triangulated category of motives $DM_k$, the group $CH^n_{\mathrm{alg}}(X)$ of codimension $n$ algebraic cycles of $X$, algebraically equivalent to zero, modulo rational equivalence, $1\leq n \leq d$. Namely, for any regular homomorphism $\psi$ (in the sense of Samuel) defined on $CH^n_{\mathrm{alg}}(X)$, we construct $M^n_{\psi}(X)\in DM_k$, which is a reasonable approximation, with respect to the slice filtration in $DM_k$, of the motive of $X$, $M(X)$; and a map $z_\psi : M^n_{\psi}(X)\rightarrow M(X)$ in $DM_k$, which computes the kernel of $\psi$. We construct as well a map, $z_{\mathrm{ab}}^n: M^n_{\mathrm{ab}}(X) \rightarrow M(X)$ having analogue properties but which instead computes the subgroup $CH^n_{\mathrm{ab}}(X)\subseteq CH^n_{\mathrm{alg}}(X)$ of algebraic cycles abelian equivalent to zero (in the sense of Samuel).