The generalized Hodge and Bloch conjectures are equivalent for general complete intersections

Voisin, Claire

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 $X_{\sigma }$ 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 $3$).

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 $3$).

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