Chow rings and decomposition theorems for K3 surfaces and Calabi–Yau hypersurfaces

The decomposition theorem for smooth projective morphisms $π : X \to B$ says that $R π_{*} ℚ$ decomposes as $\oplus R^{i} π_{*} ℚ [- i]$ . 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 $K 3$ 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 $K 3$ surface $S$ . We give two proofs of this result, the first one involving $K$ –autocorrespondences of $K 3$ 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

References

Publication

Received: 12 August 2011

Accepted: 4 December 2011

Published: 20 March 2012

Proposed: Lothar Göttsche

Seconded: Richard Thomas, Gang Tian

Authors

Claire Voisin
CNRS Institut de Mathématiques de Jussieu Case 247 4 Place Jussieu 75005 Paris France
http://www.math.jussieu.fr/~voisin/

Volume 16, issue 1 (2012)

Claire Voisin

Abstract

Keywords

Mathematical Subject Classification 2010

References

Publication

Authors