#### DOCUMENTA MATHEMATICA, Vol. 18 (2013), 1521-1553

Charles Vial

Algebraic Cycles and Fibrations

Let $f:X \rightarrow B$ 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 $B$.

2010 Mathematics Subject Classification: 14C15, 14C25, 14C05, 14D99

Keywords and Phrases: Algebraic cycles, Chow groups, quadric bundles, motives, Chow--Künneth decomposition.

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