Two coniveau filtrations and algebraic equivalence over finite fields

TL;DR

This paper extends coniveau filtration theory to the $mbda$-adic setting, shows their difference over algebraically closed fields of characteristic not 2, and explores implications for algebraic cycles and unramified cohomology over finite fields.

Contribution

It generalizes coniveau filtrations to the $mbda$-adic context, demonstrates their divergence over certain fields, and applies these results to algebraic cycles and cohomology over finite fields.

Findings

The two coniveau filtrations differ over algebraically closed fields of characteristic not 2.

Equality of filtrations over finite fields impacts algebraic equivalence of cycles.

The third unramified cohomology group vanishes for many rationally chain connected threefolds over finite fields.

Abstract

We extend the basic theory of the coniveau and strong coniveau filtrations to the $\ell$-adic setting. By adapting the examples of Benoist--Ottem to the $\ell$-adic context, we show that the two filtrations differ over any algebraically closed field of characteristic not $2$. When the base field $\mathbb{F}$ is finite, we show that the equality of the two filtrations over the algebraic closure $\overline{\mathbb{F}}$ has some consequences for algebraic equivalence for codimension-$2$ cycles over $\mathbb{F}$. As an application, we prove that the third unramified cohomology group $H^{3}_{\text{nr}}(X,\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell})$ vanishes for a large class of rationally chain connected threefolds $X$ over $\mathbb{F}$, confirming a conjecture of Colliot-Th\'el\`ene and Kahn.