Chow rings and decomposition theorems for families of $K\!3$ surfaces and Calabi–Yau hypersurfaces

Abstract

The decomposition theorem for smooth projective morphisms $\pi :\mathcal{X}\to B$ says that $R{\pi }_{\ast }\mathbb{Q}$ decomposes as $\oplus R^i{\pi }_{\ast }\mathbb{Q}\left[-i\right]$. We describe simple examples where it is not possible to have such a decomposition compatible with cup product, even after restriction to Zariski dense open sets of $B$. We prove however that this is always possible for families of $K3$ surfaces (after shrinking the base), and show how this result relates to a result by Beauville and the author [J. Algebraic Geom. 13 (2004) 417–426] on the Chow ring of a $K3$ surface $S$. We give two proofs of this result, the first one involving $K$–autocorrespondences of $K3$ surfaces, seen as analogues of isogenies of abelian varieties, the second one involving a certain decomposition of the small diagonal in $S^3$ obtained by Beauville and the author. We also prove an analogue of such a decomposition of the small diagonal in $X^3$ for Calabi–Yau hypersurfaces $X$ in ${\mathbb{P}}^n$, which in turn provides strong restrictions on their Chow ring.