Mathematics – Algebraic Geometry
Scientific paper
2010-10-10
Mathematics
Algebraic Geometry
46 pages; a typo corrected
Scientific paper
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-Th\'el\`ene, 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 vari\'et\'e propre et lisse sur un corps de nombres k. Des conjectures sur l'image du groupe de Chow des z\'ero-cycles de X dans le produit des m\^emes groupes sur tous les compl\'et\'es de k ont \'et\'e propos\'ees par Colliot-Th\'el\`ene, Kato et Saito. Nous d\'emontrons ces conjectures pour l'espace total de fibrations en vari\'et\'es rationnellement connexes v\'erifiant l'approximation faible, au-dessus de courbes dont le groupe de Tate-Shafarevich est fini, sous une hypoth\`ese d'ab\'elianit\'e sur les fibres singuli\`eres.
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