Algebraic cycles and fibrations

Abstract

Let $f:X\to B$ 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 fibres 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 Chow motive of $X$ is «built» from the motives of varieties of dimension less than the dimension of $B$.