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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

The decomposition theorem for smooth projective morphisms π: X B says that Rπ decomposes as Riπ[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 [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 S3 obtained by Beauville and the author. We also prove an analogue of such a decomposition of the small diagonal in X3 for Calabi–Yau hypersurfaces X in n, which in turn provides strong restrictions on their Chow ring.

decomposition theorem, Chow ring, decomposition of the small diagonal
Mathematical Subject Classification 2010
Primary: 14C15, 14C30, 14D99
Received: 12 August 2011
Accepted: 4 December 2011
Published: 20 March 2012
Proposed: Lothar Göttsche
Seconded: Richard Thomas, Gang Tian
Claire Voisin
Institut de Mathématiques de Jussieu
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