The generalized Hodge and Bloch conjectures are equivalent for general complete intersections
[Les conjectures de Hodge et de Bloch généralisées sont équivalentes pour les intersections complètes générales]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 3, pp. 449-475.

Nous montrons la conjecture de Bloch pour les surfaces avec pg=0 obtenues comme lieux des zéros Xσ d’une section σ d’un fibré vectoriel très ample sur une variété X à groupes de Chow « triviaux ». Nous obtenons un résultat similaire en présence d’une action d’un groupe fini, montrant que si un projecteur du groupe agit comme 0 sur les 2-formes holomorphes de Xσ, il agit comme 0 sur les 0-cycles de degré 0 de Xσ. En dimension supérieure, nous obtenons un résultat similaire mais conditionnel montrant que la conjecture de Hodge généralisée pour Xσ générale entraîne la conjecture de Bloch généralisée pour tout Xσ lisse, en supposant satisfaite la conjecture de Lefschetz standard (cette dernière hypothèse n’étant pas nécessaire en dimension 3).

We prove that Bloch’s conjecture is true for surfaces with pg=0 obtained as 0-sets Xσ of a section σ of a very ample vector bundle on a variety X with “trivial” Chow groups. We get a similar result in presence of a finite group action, showing that if a projector of the group acts as 0 on holomorphic 2-forms of Xσ, then it acts as 0 on 0-cycles of degree 0 of Xσ. In higher dimension, we also prove a similar but conditional result showing that the generalized Hodge conjecture for general Xσ implies the generalized Bloch conjecture for any smooth Xσ, assuming the Lefschetz standard conjecture (the last hypothesis is not needed in dimension 3).

DOI : 10.24033/asens.2193
Classification : 14C25, 14C30
Keywords: algebraic cycles, Bloch conjecture, generalized Hodge conjecture
Mot clés : cycles algébriques, conjecture de Bloch, conjecture de Hodge généralisée
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Voisin, Claire. The generalized Hodge and Bloch conjectures are equivalent for general complete intersections. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 3, pp. 449-475. doi : 10.24033/asens.2193. https://www.numdam.org/articles/10.24033/asens.2193/

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