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

Claire Voisin

Annales scientifiques de l'École Normale Supérieure (2013)

  • Volume: 46, Issue: 3, page 449-475
  • ISSN: 0012-9593

Abstract

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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 ).

How to cite

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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 46.3 (2013): 449-475. <http://eudml.org/doc/272244>.

@article{Voisin2013,
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 $3$).},
author = {Voisin, Claire},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {algebraic cycles; Bloch conjecture; generalized Hodge conjecture},
language = {eng},
number = {3},
pages = {449-475},
publisher = {Société mathématique de France},
title = {The generalized Hodge and Bloch conjectures are equivalent for general complete intersections},
url = {http://eudml.org/doc/272244},
volume = {46},
year = {2013},
}

TY - JOUR
AU - Voisin, Claire
TI - The generalized Hodge and Bloch conjectures are equivalent for general complete intersections
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2013
PB - Société mathématique de France
VL - 46
IS - 3
SP - 449
EP - 475
AB - 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 $3$).
LA - eng
KW - algebraic cycles; Bloch conjecture; generalized Hodge conjecture
UR - http://eudml.org/doc/272244
ER -

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