Motives and representability of algebraic cycles on threefolds over a field

TL;DR

This paper explores the relationship between algebraic cycles and the finite-dimensionality of motives on threefolds, providing new decompositions and proving finite-dimensionality for certain classes like Fano threefolds.

Contribution

It introduces a motive decomposition for non-singular projective threefolds with representable zero-cycles and proves finite-dimensionality for Fano threefolds and other classes.

Findings

Decomposition of motives into Lefschetz and Picard parts.

Finite-dimensionality of motives for Fano threefolds.

Representability of zero-cycles on fibered threefolds.

Abstract

We study links between algebraic cycles on threefolds and finite-dimensionality of their motives with coefficients in Q. We decompose the motive of a non-singular projective threefold X with representable algebraic part of CH_0(X) into Lefschetz motives and the Picard motive of a certain abelian variety, isogenous to the corresponding intermediate Jacobian J^2(X) when the ground field is C. In particular, it implies motivic finite-dimensionality of Fano threefolds over a field. We also prove representability of zero-cycles on several classes of threefolds fibered by surfaces with algebraic H^2. This gives another new examples of three-dimensional varieties whose motives are finite-dimensional.