Geometry & Topology

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

Claire Voisin

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The decomposition theorem for smooth projective morphisms π:XB 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.

Article information

Geom. Topol., Volume 16, Number 1 (2012), 433-473.

Received: 12 August 2011
Accepted: 4 December 2011
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 14C15: (Equivariant) Chow groups and rings; motives 14C30: Transcendental methods, Hodge theory [See also 14D07, 32G20, 32J25, 32S35], Hodge conjecture 14D99: None of the above, but in this section

decomposition theorem Chow ring decomposition of the small diagonal


Voisin, Claire. Chow rings and decomposition theorems for families of $K\!3$ surfaces and Calabi–Yau hypersurfaces. Geom. Topol. 16 (2012), no. 1, 433--473. doi:10.2140/gt.2012.16.433.

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