Chow-Kuenneth decomposition for 3- and 4-folds fibred by varieties with small Chow group of zero-cycles

TL;DR

This paper proves that certain 3- and 4-dimensional algebraic varieties fibred over surfaces with trivial zero-cycle Chow groups have a self-dual Murre decomposition, satisfying key conjectures and providing new examples of varieties with special motivic properties.

Contribution

It establishes a self-dual Chow--Kuenneth decomposition for specific fibred varieties, confirming Murre's conjectures and the motivic Lefschetz conjecture in these cases.

Findings

Proves Murre decomposition for certain fibred varieties.

Confirms motivic Lefschetz and standard conjectures for these varieties.

Provides new examples of varieties with finite-dimensional motives and special cohomological properties.

Abstract

Let $k$ be a field and let $\Omega$ be a universal domain over $k$. Let $f:X \r S$ be a dominant morphism defined over $k$ from a smooth projective variety $X$ to a smooth projective variety $S$ of dimension $\leq 2$ such that the general fibre of $f_\Omega$ has trivial Chow group of zero-cycles. For example, $X$ could be the total space of a two-dimensional family of varieties whose general member is rationally connected. Suppose that $X$ has dimension $\leq 4$. Then we prove that $X$ has a self-dual Murre decomposition, i.e. that $X$ has a self-dual Chow--Kuenneth decomposition which satisfies Murre's conjectures (B) and (D). Moreover we prove that the motivic Lefschetz conjecture holds for $X$ and hence so does the Lefschetz standard conjecture. We also give new examples of threefolds of general type which are Kimura finite-dimensional, new examples of fourfolds of general type having a self-dual Murre decomposition, as well as new examples of varieties with finite degree three unramified cohomology.