Algebraic cycles and fibrations

TL;DR

This paper investigates how the Chow groups of a variety relate to those of its base and fibers in a fibration, with applications to quadric bundles and motives in algebraic geometry.

Contribution

It provides new insights into the structure of Chow groups and motives for fibrations, especially flat quadric fibrations, connecting the geometry of total space, base, and fibers.

Findings

Chow groups of total space relate to those of base and fibers.

Motives of flat quadric fibrations are constructed from lower-dimensional motives.

Application to understanding algebraic cycles in quadric bundles.

Abstract

Let $f : X -> B$ be a projective surjective morphism between quasi-projective varieties. The goal of this paper is the study of the Chow groups of $X$ in terms of the Chow groups of $B$ and of the fibers of $f$. One of the applications concerns quadric bundles. When $X$ and $B$ are smooth projective and when $f$ is a flat quadric fibration, we show that the motive of $X$ is "built" from the motives of varieties of dimension less than the dimension of $B$.