Chow groups of smooth varieties fibred by quadrics

TL;DR

This paper extends the projective bundle formula to certain fibred varieties, enabling the construction of Chow-Künneth decompositions that satisfy Murre's conjectures for specific classes of complex varieties.

Contribution

It proves an analogue of the projective bundle formula for Chow groups in fibred varieties and constructs Chow-Künneth decompositions satisfying Murre's conjectures in new cases.

Findings

Established a projective bundle formula analogue for specific Chow groups.

Constructed Chow-Künneth decompositions satisfying Murre's conjectures for fibred varieties.

Proved Murre's conjectures for varieties fibred over surfaces by quadrics or certain complete intersections.

Abstract

Let $f : X \rightarrow B$ 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 $j \leq l$. Then we prove an analogue of the projective bundle formula for $CH_i(X)$ for $i \leq l$. 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-K\"unneth 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 $(2,2)$.