Sur le groupe de Chow de codimension deux des vari\'et\'es sur les corps finis

TL;DR

This paper constructs an example of a rational variety over a finite field where certain cohomological invariants are nontrivial, revealing new insights into the Chow group structure in positive characteristic.

Contribution

It provides the first explicit example over finite fields showing nontrivial unramified cohomology and non-surjectivity of the Chow group map, advancing understanding of algebraic cycles in positive characteristic.

Findings

Nontrivial unramified cohomology group H^3_nr(X, Z/2)

Non-surjectivity of the Chow group map CH^2(X) to CH^2(ar X)^G

Explicit example over finite fields demonstrating these properties

Abstract

Using the construction of Colliot-Th\'el\`ene and Ojanguren, we exhibit an example of a smooth projective geometrically rational variety X defined over a finite field F_p with an algebraic closure \bar F_p and the absolute Galois group G, such that the group H^3_nr(X, Z/2) is nonzero and the map CH^2(X)\to CH^2(\bar X)^G is not surjective.