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

TL;DR

This paper investigates the decomposition theorem for families of K3 surfaces and Calabi-Yau hypersurfaces, demonstrating compatibility with the cup-product after base changes and exploring implications for their Chow rings.

Contribution

It proves that the decomposition theorem can be made compatible with cup-product for K3 families and extends similar results to Calabi-Yau hypersurfaces, linking to Chow ring structures.

Findings

Decomposition compatible with cup-product for K3 families after base change

Decomposition of the small diagonal in Calabi-Yau hypersurfaces

Restrictions on Chow rings of Calabi-Yau hypersurfaces

Abstract

The decomposition theorem for smooth projective morphisms $\pi:\mathcal{X}\rightarrow B$ says that $R\pi_*\mathbb{Q}$ decomposes as $\oplus R^i\pi_*\mathbb{Q}[-i]$. 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.