Research Papers

Regular homomorphisms and mixed motives

I. Hernandez^a, P. Pelaez<sup>b

a</sup> Instituto de Matemáticas, Ciudad Universitaria, UNAM, DF 04510, México
^b Chebyshev Laboratory, St. Petersburg State University, 14th Line V. O., 29B, Saint Petersburg 199178 Russia

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 similar 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).

Keywords: Abelian equivalence, Abel–Jacobi equivalence, chow groups, incidence equivalence, mixed motives, regular homomorphisms, slice filtration.

Received: 19.10.2025

Language: English