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On the Chow Ring of Cynk–Hulek Calabi–Yau Varieties and Schreieder Varieties

Part of: Arithmetic problems. Diophantine geometry Surfaces and higher-dimensional varieties Cycles and subschemes

Published online by Cambridge University Press:  03 September 2019

Robert Laterveer  and
Charles Vial
Robert Laterveer
Affiliation:
Institut de Recherche Mathématique Avancée, CNRS – Université de Strasbourg, 7 Rue René Descartes, 67084 Strasbourg CEDEX, France Email: robert.laterveer@math.unistra.fr
Charles Vial
Affiliation:
Universität Bielefeld, Fakultät für Mathematik, Postfach 10031, 33501 Bielefeld, Germany Email: vial@math.uni-bielefeld.de
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Abstract

This note is about certain locally complete families of Calabi–Yau varieties constructed by Cynk and Hulek, and certain varieties constructed by Schreieder. We prove that the cycle class map on the Chow ring of powers of these varieties admits a section, and that these varieties admit a multiplicative self-dual Chow–Künneth decomposition. As a consequence of both results, we prove that the subring of the Chow ring generated by divisors, Chern classes, and intersections of two cycles of positive codimension injects into cohomology via the cycle class map. We also prove that the small diagonal of Schreieder surfaces admits a decomposition similar to that of K3 surfaces. As a by-product of our main result, we verify a conjecture of Voisin concerning zero-cycles on the self-product of Cynk–Hulek Calabi–Yau varieties, and in the odd-dimensional case we verify a conjecture of Voevodsky concerning smash-equivalence. Finally, in positive characteristic, we show that the supersingular Cynk–Hulek Calabi–Yau varieties provide examples of Calabi–Yau varieties with “degenerate” motive.

Keywords

algebraic cycleChow ringmotiveBloch–Beilinson filtrationmultiplicative Chow–Künneth decompositionCalabi–Yau varietysupersingular variety

MSC classification

Primary: 14C15: (Equivariant) Chow groups and rings; motives
Secondary: 14C25: Algebraic cycles 14C30: Transcendental methods, Hodge theory , Hodge conjecture 14J32: Calabi-Yau manifolds 14G17: Positive characteristic ground fields
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 2 , April 2020 , pp. 505 - 536
Copyright
© Canadian Mathematical Society 2019

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