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 $D{M}_k$, the group $C{H}_{\mathrm{alg}}^n(X)$ of codimension $n$ algebraic cycles of $X$, algebraically equivalent to zero, modulo rational equivalence, $1\le n\le d$. Namely, for any regular homomorphism $\ddot{I}ˆ$ (in the sense of Samuel) defined on $C{H}_{\mathrm{alg}}^n(X)$, we construct $M_{\ddot{I}}^nˆ(X)\in D{M}_k$, which is a reasonable approximation, with respect to the slice filtration in $D{M}_k$, of the motive of $X$, $M(X)$; and a map $z_{\ddot{I}}ˆ:M_{\ddot{I}}^nˆ(X)\to M(X)$ in $D{M}_k$, which computes the kernel of $\ddot{I}ˆ$. We construct as well a map, $z_{\mathrm{ab}}^n:M_{\mathrm{ab}}^n(X)\to M(X)$ having analogue properties but which instead computes the subgroup $C{H}_{\mathrm{ab}}^n(X)\subseteq C{H}_{\mathrm{alg}}^n(X)$ of algebraic cycles abelian equivalent to zero (in the sense of Samuel).

18 pages