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

The decomposition theorem for smooth projective morphisms W X ! B says that R Q decomposes as L R QŒ 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 [2] 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 obtained in [2]. We also prove an analogue of such a decomposition of the small diagonal in X 3 for Calabi–Yau hypersurfaces X in P n , which in turn provides strong restrictions on their Chow ring.