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.

中文翻译：

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在 CYNK-HULEK CALABI-YAU 品种和 SCHREIEDER 品种的周环上

本注释是关于由 Cynk 和 Hulek 构建的某些本地完整的 Calabi-Yau 品种家族，以及由 Schreieder 构建的某些品种。我们证明这些变体的幂的 Chow 环上的循环类图允许一个截面，并且这些变体允许乘法自对偶 Chow-Kuenneth 分解。作为这两个结果的结果，我们证明了由除数、陈类和两个正余维循环的交集生成的 Chow 环的子环通过循环类映射注入上同调。我们还证明了 Schreieder 曲面的小对角线允许类似于 K3 曲面的分解。作为我们主要结果的副产品，我们验证了关于 Cynk-Hulek Calabi-Yau 品种自积零循环的 Voisin 猜想，在奇维情况下，我们验证了 Voevodsky 关于 smash-equivalence 的猜想。最后，在积极特征方面，我们表明超奇异 Cynk-Hulek Calabi-Yau 变种提供了具有“退化”动机的 Calabi-Yau 变种的例子。