Chow rings and decomposition theorems for families of K3 surfaces and Calabi-Yau hypersurfaces

By Claire Voisin. 2011

49 citations recorded by the source corpus.

The decomposition theorem for smooth projective morphisms W X ! B says that R Q decomposes as L R QŒ 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 [2] 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 S obtained in [2]. 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.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4abb7c79fea860c0d7cedcdc26521e8d3d83cdd5

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In Geometry and Topology

1. M. Shen
2. Charles Vial
3. L. Fu
4. Zhiyu Tian
5. Claire Voisin
6. G. Farkas
7. M. Aprodu
8. Stefan Schreieder

Citations in the corpus

• The Fourier Transform for Certain HyperKähler Fourfolds (2013, background): “This is proved for smooth Calabi–Yau hypersufaces by Voisin [55] and for smooth Calabi–Yau complete intersections by Fu [22].”
• THE MOTIVE OF THE HILBERT CUBE $X^{[3]}$ (2015, background): “[14] C. Voisin, The generalized Hodge and Bloch conjectures are equivalent for general complete intersections, Ann.”
• Motivic hyper-Kähler resolution conjecture, I : Generalized Kummer varieties (2016, background): “Voisin [59] shows that, although this isomorphism cannot in general be made compatible with the product structures on both sides, not even after shrinking B to a Zariski open subset, it can be made so if is a smooth family of projective K3 surfaces.”
• Chow Rings, Decomposition of the Diagonal, and the Topology of Families (2014, background): “1 Calabi–Yau hypersurfaces In the case of smooth Calabi–Yau hypersurfaces X in projective space P, that is, hypersurfaces of degree (n+ 1) in P, we have the following result proved in [110], which partially generalizes Theorem 5.”