Chow groups of smooth varieties fibred by quadrics

Abstract

Let $f:X\to B$ be a proper flat dominant morphism between two smooth quasi-projective complex varieties $X$ and $B$. Assume that there exists an integer $l$ such that all closed fibres $X_b$ of $f$ satisfy $CH_j(X_b) = \Q$ for all $j\le l$. Then we prove an analogue of the projective bundle formula for $C{H}_i(X)$ for $i\le l$. When $B$ is a surface, $X$ is projective and $l=\lfloor \frac{\dim X-3}{2}\rfloor$, this makes it possible to construct a Chow-Künneth decomposition for $X$ that satisfies Murre's conjectures. For instance we prove Murre's conjectures for complex smooth projective varieties $X$ fibred over a surface (via a flat morphism) by quadrics, or by complete intersections of dimension 4 of bidegree $(2,2)$.

20 pages