On the Chow ring of Cynk-Hulek Calabi-Yau varieties and Schreieder varieties

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-Kuenneth 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.

to appear in the Canadian Journal of Mathematics

References in corpus (3)

• On the splitting of the Bloch-Beilinson filtration
• Algebraic cycles and Todorov surfaces
• Construction and examples of higher-dimensional modular Calabi-Yau manifolds

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• On the Chow ring of Fano varieties on the Fatighenti-Mongardi list
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• Algebraic cycles on Todorov surfaces of type $(2,12)$
• Zero-cycles on self-products of varieties: some elementary examples verifying Voisin's conjecture
• Zero-cycles on Garbagnati surfaces
• On the Chow ring of certain Fano fourfolds
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• Some new Fano varieties with a multiplicative Chow-Künneth decomposition
• Algebraic cycles and Lehn-Lehn-Sorger-van Straten eightfolds