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

TL;DR

This paper investigates the Chow rings and motives of Cynk-Hulek Calabi-Yau and Schreieder varieties, establishing new structural decompositions, verifying conjectures, and exploring properties in positive characteristic.

Contribution

It proves the existence of a section for the cycle class map, constructs a multiplicative self-dual Chow-Kuenneth decomposition, and verifies conjectures by Voisin and Voevodsky for these varieties.

Findings

Cycle class map admits a section for these varieties.

Varieties admit a multiplicative self-dual Chow-Kuenneth decomposition.

Verification of Voisin's conjecture and Voevodsky's smash-equivalence in specific cases.

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.