Torsion algebraic cycles and etale cobordism

TL;DR

This paper demonstrates that algebraic cycles' cohomology classes in positive characteristic are obstructed by topological factors, extending classical results through étale cobordism and confirming longstanding examples in a new setting.

Contribution

It proves the cycle class map factors through étale cobordism in positive characteristic, revealing topological obstructions to algebraicity of cohomology classes.

Findings

Cycle class map factors through étale cobordism.

Topological obstructions prevent certain cohomology classes from being algebraic.

Examples by Atiyah, Hirzebruch, and Totaro apply in positive characteristic.

Abstract

Following an idea of Totaro, we prove that the classical integral cycle class map from algebraic cycles to \'etale cohomology factors through a quotient of $\ell$-adic \'etale cobordism over an algebraically closed field of positive characteristic. This shows that there is a strong topological obstruction for cohomology classes to be algebraic and that examples of Atiyah, Hirzebruch and Totaro also work in positive characteristic.