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

Abstract

Let $k$ be a field and let $\Omega$ be a universal domain over $k$. Let $f:X\mathring{S}$ be a dominant morphism defined over $k$ from a smooth projective variety $X$ to a smooth projective variety $S$ of dimension $\le 2$ 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 $\le 4$. 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.

27 pages

References in corpus (2)

• Projectors on the intermediate algebraic Jacobians
• Cycles de codimension 2 et H^3 non ramifié pour les variétés sur les corps finis