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 p g =0 obtenues comme lieux des zéros X σ 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 X σ , il agit comme 0 sur les 0-cycles de degré 0 de X σ . En dimension supérieure, nous obtenons un résultat similaire mais conditionnel montrant que la conjecture de Hodge généralisée pour X σ générale entraîne la conjecture de Bloch généralisée pour tout X σ 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 p g =0 obtained as 0-sets X σ 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 X σ , then it acts as 0 on 0-cycles of degree 0 of X σ . In higher dimension, we also prove a similar but conditional result showing that the generalized Hodge conjecture for general X σ implies the generalized Bloch conjecture for any smooth X σ , 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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     title = {The generalized {Hodge} and {Bloch} conjectures are equivalent for general complete intersections},
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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. http://archive.numdam.org/articles/10.24033/asens.2193/

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