#### Volume 16, issue 1 (2012)

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

### Claire Voisin

Geometry & Topology 16 (2012) 433–473
##### Abstract

The decomposition theorem for smooth projective morphisms $\pi :\mathsc{X}\to B$ says that $R{\pi }_{\ast }ℚ$ decomposes as $\oplus {R}^{i}{\pi }_{\ast }ℚ\left[-i\right]$. 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 ${ℙ}^{n}$, which in turn provides strong restrictions on their Chow ring.

##### Keywords
decomposition theorem, Chow ring, decomposition of the small diagonal
##### Mathematical Subject Classification 2010
Primary: 14C15, 14C30, 14D99