Steenrod operations and algebraic classes

Abstract

Based on a relative Wu theorem in étale cohomology, we study the compatibility of Steenrod operations on Chow groups and on étale cohomology. Using the resulting obstructions to algebraicity, we construct new examples of non-algebraic cohomology classes over various fields ($\mathbb{C}$, $\mathbb{R}$, ${\overline{\mathbb{F}}}_p$, ${\mathbb{F}}_q$). We also use Steenrod operations to study the mod $2$ cohomology classes of a compact ${\mathcal{C}}^{\infty }$ manifold $M$ that are algebraizable, i.e. algebraic on some real algebraic model of $M$. We give new examples of algebraizable and non-algebraizable classes, answering questions of Benedetti, Dedò and Kucharz.

arXiv comment33 pages, final version

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