On Chow motives of 3-folds

Abstract:

Let $k$ be a field of characteristic zero. For every smooth, projective $k$-variety $Y$ of dimension $n$ which admits a connected, proper morphism $f: Y \to S$ of relative dimension one, we construct idempotent correspondences (projectors) $\pi _{ij}(Y) \in CH^{n}(Y \times Y,\mathbb {Q})$ generalizing a construction of Murre. If $n=3$ and the transcendental cohomology group $H^{2}_{\text {tr}}(Y)$ has the property that $H^{2}_{\text {tr}}(Y,\mathbb {C})=f^{*}H^{2}_{\text {tr}}(S,\mathbb {C})+ {\text {Im}}(f^{*}H^{1}(S,\mathbb {C}) \otimes H^{1}(Y,\mathbb {C}) \to H^{2}_{\text {tr}}(Y,\mathbb {C}))$, then we can construct a projector $\pi _{2}(Y)$ which lifts the second Künneth component of the diagonal of $Y$. Using this we prove that many smooth projective 3-folds $X$ over $k$ admit a Chow-Künneth decomposition $\Delta =p_{0}+...+p_{6}$ of the diagonal in $CH^{3}(X \times X,{\mathbb {Q}})$.