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

Let $k$ be a field and let $\mathrm{\Omega }$ be a universal domain over $k$. Let $f:X \r S$beadominantmorphismdefinedover$k$fromasmoothprojectivevariety$X$toasmoothprojectivevariety$S$ofdimension$\leq 2$ such that the general fibre of $f_{\mathrm{\Omega }}$ 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 $\leq 4$\mathrm{.}Thenweprovethat$X$hasaself-dualMurredecomposition,i\mathrm{.}e\mathrm{.}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.