Steenrod operations and algebraic classes

TL;DR

The paper develops a relative Wu theorem in étale cohomology to compare Steenrod operations on Chow groups and étale cohomology, yielding obstructions to algebraicity that extend topological results to non-closed fields. It constructs numerous non-algebraic cohomology classes across complex, finite field, and real settings, including new counterexamples to the integral Tate and integral Hodge conjectures, and it develops algebraizability criteria for cohomology classes of smooth manifolds. It further connects these obstructions to complex cobordism and unramified cohomology, and provides a detailed study of the stability of algebraizable classes under Steenrod operations, along with explicit algebraizable and non-algebraizable examples in the real setting.

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.

Theorems & Definitions (77)

• Theorem 1.1: Theorem \ref{['compatibility']}
• Theorem 1.2: Theorem \ref{['th1b']}
• Theorem 1.3: Theorem \ref{['th2b']}
• Theorem 1.4: Theorem \ref{['finitethm']}
• Theorem 1.5: Theorem \ref{['threal']}
• Theorem 1.6: Theorem \ref{['propnonalgebraizable']}
• Theorem 1.7: Theorem \ref{['cexSWsub']}
• Lemma 2.2
• proof
• Lemma 2.3