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]

Voisin, Claire

Annales scientifiques de l'École Normale Supérieure, Tome 46 (2013), p. 449-475 / Harvested from Numdam

Résumé

Nous montrons la conjecture de Bloch pour les surfaces avec $p_g=0$ obtenues comme lieux des zéros $X_\sigma $ d’une section $\sigma $ 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_\sigma $, il agit comme $0$ sur les $0$-cycles de degré $0$ de $X_\sigma $. En dimension supérieure, nous obtenons un résultat similaire mais conditionnel montrant que la conjecture de Hodge généralisée pour $X_\sigma $ générale entraîne la conjecture de Bloch généralisée pour tout $X_\sigma $ lisse, en supposant satisfaite la conjecture de Lefschetz standard (cette dernière hypothèse n’étant pas nécessaire en dimension $3$).

Abstract

We prove that Bloch’s conjecture is true for surfaces with $p_g=0$ obtained as $0$-sets $X_\sigma $ of a section $\sigma $ 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_\sigma $, then it acts as $0$ on $0$-cycles of degree $0$ of $X_\sigma $. In higher dimension, we also prove a similar but conditional result showing that the generalized Hodge conjecture for general $X_\sigma $ implies the generalized Bloch conjecture for any smooth $X_\sigma $, assuming the Lefschetz standard conjecture (the last hypothesis is not needed in dimension $3$).

Détail

Publié le : 2013-01-01
DOI : https://doi.org/10.24033/asens.2193
Classification:  14C25,  14C30
Mots clés: cycles algébriques, conjecture de Bloch, conjecture de Hodge généralisée

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