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arXiv · 1010.1883

Zéro-cycles sur les fibrations au-dessus d'une courbe de genre quelconque

Olivier Wittenberg
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Abstract

Let X be a smooth and proper variety over a number field k. Conjectures on the image of the Chow group of zero-cycles of X in the product of the corresponding groups over all completions of k were put forward by Colliot-Thélène, Kato and Saito. We prove these conjectures for the total space of fibrations, over curves with finite Tate-Shafarevich group, into rationally connected varieties which satisfy weak approximation, under an abelianness assumption on the singular fibers. ---- Soit X une variété propre et lisse sur un corps de nombres k. Des conjectures sur l'image du groupe de Chow des zéro-cycles de X dans le produit des mêmes groupes sur tous les complétés de k ont été proposées par Colliot-Thélène, Kato et Saito. Nous démontrons ces conjectures pour l'espace total de fibrations en variétés rationnellement connexes vérifiant l'approximation faible, au-dessus de courbes dont le groupe de Tate-Shafarevich est fini, sous une hypothèse d'abélianité sur les fibres singulières.

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Olivier Wittenberg. 2012-05-31. Zéro-cycles sur les fibrations au-dessus d'une courbe de genre quelconque. https://doi.org/10.1215/00127094-1699441

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