Geometry & Topology

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

Claire Voisin

Abstract

The decomposition theorem for smooth projective morphisms $π:X→B$ says that $Rπ∗ℚ$ decomposes as $⊕Riπ∗ℚ[−i]$. 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 [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 $ℙn$, which in turn provides strong restrictions on their Chow ring.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732386

Digital Object Identifier
doi:10.2140/gt.2012.16.433

Mathematical Reviews number (MathSciNet)
MR2916291

Zentralblatt MATH identifier
1253.14005

Citation

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. https://projecteuclid.org/euclid.gt/1513732386

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