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

Published online by Cambridge University Press:  04 February 2013

Jean-Louis Colliot-Thélène  and
Bruno Kahn
Jean-Louis Colliot-Thélène
Affiliation:
C.N.R.S., Université Paris Sud, UMR 8628, Mathématiques, Bâtiment 425, 91405 Orsay CedexFrancejlct@math.u-psud.fr
Bruno Kahn
Affiliation:
Institut de Mathématiques de Jussieu, UMR 7586, Case 247, 4 place Jussieu, 75252 Paris Cedex 05, Francekahn@math.jussieu.fr
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Abstract

Let X be a smooth projective variety over a finite field . 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 (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.

Keywords

unramified cohomologyalgebraic cyclesfinite fieldslocalglobal principle19E1514G1514G2514C2514C35

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Type
Research Article
Information
Journal of K-Theory , Volume 11 , Issue 1 , February 2013 , pp. 1 - 53
Copyright
Copyright © ISOPP 2013

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