Chow rings and decomposition theorems for families of K3 surfaces and Calabi-Yau hypersurfaces

The decomposition theorem for smooth projective morphisms $\pi :\mathcal{X}\to B$ says that $R{\pi }_{\ast }\mathbb{Q}$ decomposes as $\oplus R^i{\pi }_{\ast }\mathbb{Q}[-i]$. 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 on the Chow ring of $K3$ surfaces $S$. We give two proofs of this result, the second one involving a certain decomposition of the small diagonal in $S^3$ also proved by Beauville and the author}. We 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.