Geometry & Topology
- Geom. Topol.
- Volume 16, Number 1 (2012), 433-473.
Chow rings and decomposition theorems for families of $K\!3$ surfaces and Calabi–Yau hypersurfaces
The decomposition theorem for smooth projective morphisms says that decomposes as . 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 . We prove however that this is always possible for families of 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 surface . We give two proofs of this result, the first one involving –autocorrespondences of surfaces, seen as analogues of isogenies of abelian varieties, the second one involving a certain decomposition of the small diagonal in obtained by Beauville and the author. We also prove an analogue of such a decomposition of the small diagonal in for Calabi–Yau hypersurfaces in , which in turn provides strong restrictions on their Chow ring.
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
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