Cycles de codimension 2 et H3 non ramifié pour les variétés sur les corps finis

Author:

Colliot-Thélène Jean-Louis,Kahn Bruno

Abstract

AbstractLet X be a smooth projective variety over a finite field $\mathbb{F}$. We discuss the unramified cohomology group H3nr(X, ℚ/ℤ(2)). Several conjectures put together imply that this group is finite. For certain classes of threefolds, H3nr(X, ℚ/ℤ(2)) actually vanishes. It is an open question whether this holds for arbitrary threefolds. For a threefold X equipped with a fibration onto a curve C, the generic fibre of which is a smooth projective surface V over the global field $\mathbb{F}$(C), the vanishing of H3nr(X, ℚ/ℤ(2)) together with the Tate conjecture for divisors on X implies a local-global principle of Brauer–Manin type for the Chow group of zero-cycles on V. This sheds new light on work started thirty years ago.

Publisher

Cambridge University Press (CUP)

Subject

Geometry and Topology,Algebra and Number Theory

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