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

摘要

The decomposition theorem for smooth projective morphisms π: X → B says that Rπ_*Q decomposes as direct+ R~iπ_*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 [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~3 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.