Motives of uniruled 3-folds

TL;DR

This paper constructs specific algebraic projectors for uniruled 3-folds that support conjectures about the structure of their Chow groups, extending Murre's conjectures to new classes of varieties.

Contribution

It introduces a method using Mori theory to construct Murre projectors for uniruled 3-folds, advancing the understanding of their Chow groups and filtrations.

Findings

Constructed Murre projectors for uniruled 3-folds.

Supported Bloch-Beilinson conjectures on Chow groups.

Extended Murre's decomposition to new geometric contexts.

Abstract

We construct projectors in the ring of correspondences of a complex uniruled 3-fold $X$ which lift the Kuenneth components of the diagonal in singular cohomology and have other properties which were conjectured by J. Murre. Such Murre decompositions have been already obtained for curves, surfaces, abelian varieties and varieties with cell decompositions by the work of Manin, Shermenev, Beauville, Murre et.al.. In particular they define a natural filtration on the Chow groups of $X$ which was conjectured by Bloch and Beilinson. To do this we use Mori theory and construct projectors in the situation of a fiber space over a surface. These projectors may also be used in more general situations.