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localglobal principle

Journal of K-Theory (1)

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Cycles de codimension 2 et H3 non ramifié pour les variétés sur les corps finis
Jean-Louis Colliot-Thélène, Bruno Kahn
Journal:

Journal of K-Theory / Volume 11 / Issue 1 / February 2013

Published online by Cambridge University Press:

04 February 2013, pp. 1-53

Print publication:

February 2013
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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.