Steenrod operations and algebraic classes

TL;DR

This paper explores the compatibility of Steenrod operations with algebraic classes in cohomology, providing new examples of algebraic and non-algebraic classes over various fields and addressing questions about algebraizability.

Contribution

It introduces new obstructions to algebraicity using Steenrod operations and constructs examples of algebraic and non-algebraic classes, advancing understanding of algebraizability over different fields.

Findings

Constructed new examples of non-algebraic cohomology classes.

Provided examples of algebraizable and non-algebraizable classes on manifolds.

Analyzed compatibility of Steenrod operations with algebraic classes.

Abstract

Based on a relative Wu theorem in \'etale cohomology, we study the compatibility of Steenrod operations on Chow groups and on \'etale 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, Ded\`o and Kucharz.