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La conjecture de Tate entière pour les cubiques de dimension quatre

Part of:Cycles and subschemesArithmetic problems. Diophantine geometrySurfaces and higher-dimensional varieties

Published online by Cambridge University Press:  16 October 2014

François Charles and
Alena Pirutka
François Charles
Affiliation:
Département de Mathématiques, Université Paris-Sud, Bâtiment 425, 91405 Orsay cedex, France email francois.charles@math.u-psud.fr
Alena Pirutka
Affiliation:
Université de Strasbourg, IRMA – UMR 7501 du CNRS, 7 rue René Descartes, 67084 Strasbourg cedex, France email pirutka@math.unistra.fr
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Abstract

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We prove the integral Tate conjecture for cycles of codimension $2$ on smooth cubic fourfolds over an algebraic closure of a field finitely generated over its prime subfield and of characteristic different from $2$ or $3$. The proof relies on the Tate conjecture with rational coefficients, proved in that setting by the first author, and on an argument of Voisin coming from complex geometry.

Résumé

Dans ce texte, on établit une version entière de la conjecture de Tate pour les cycles de codimension $2$ sur une hypersurface cubique lisse $X$ de $\mathbb{P}^{5}$ sur une clôture algébrique d’un corps de type fini sur son sous-corps premier et de caractéristique différente de $2$ et $3$. La preuve s’appuie sur la conjecture de Tate à coefficients rationnels prouvée dans ce cas par le premier auteur et sur un argument de géométrie complexe dû à Voisin.

Keywords

conjecture de Tatecubique de dimension 4jacobienne intermédiairefonction normale

MSC classification

Primary:14C25: Algebraic cycles14G15: Finite ground fields14J70: Hypersurfaces

Information

Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 2 , February 2015, pp. 253 - 264
Copyright
© The Author(s) 2014 

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