Finite-dimensional motives

Yves André

Séminaire Bourbaki (2003-2004)

  • Volume: 46, page 115-146
  • ISSN: 0303-1179

Abstract

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It is known that the Chow groups of a projective variety are not finite-dimensional and cannot even be parametrized by an algebraic variety in general. However, S.-I. Kimura and P. O’Sullivan have (independently) conjectured that Chow motives are “finite-dimensional“in the sense that, like super vector bundles, they can be decomposed into an “even” motive (whose high exterior power vanish) and an “odd” motive (whose high symmetric powers vanish). The theory of this purely categorical notion is presented, as well as some applications in algebraic geometry.

How to cite

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André, Yves. "Motifs de dimension finie." Séminaire Bourbaki 46 (2003-2004): 115-146. <http://eudml.org/doc/252130>.

@article{André2003-2004,
abstract = {On sait que les groupes de Chow d’une variété projective ne sont pas de type fini, et ne peuvent même être paramétrés par une variété algébrique, en général. Pourtant, S.-I. Kimura et P. O’Sullivan ont conjecturé (indépendamment l’un de l’autre) que les motifs de Chow, définis en termes de correspondances algébriques modulo l’équivalence rationnelle, sont de “dimension finie”au sens où, tout comme les super-fibrés vectoriels, ils sont somme d’un facteur dont une puissance extérieure est nulle et d’un facteur dont une puissance symétrique est nulle. Je présenterai la théorie de cette notion (purement catégorique), puis ses applications en géométrie algébrique.},
author = {André, Yves},
journal = {Séminaire Bourbaki},
keywords = {Chow groups; motives; tensor categories; parity},
language = {fre},
pages = {115-146},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {Motifs de dimension finie},
url = {http://eudml.org/doc/252130},
volume = {46},
year = {2003-2004},
}

TY - JOUR
AU - André, Yves
TI - Motifs de dimension finie
JO - Séminaire Bourbaki
PY - 2003-2004
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 46
SP - 115
EP - 146
AB - On sait que les groupes de Chow d’une variété projective ne sont pas de type fini, et ne peuvent même être paramétrés par une variété algébrique, en général. Pourtant, S.-I. Kimura et P. O’Sullivan ont conjecturé (indépendamment l’un de l’autre) que les motifs de Chow, définis en termes de correspondances algébriques modulo l’équivalence rationnelle, sont de “dimension finie”au sens où, tout comme les super-fibrés vectoriels, ils sont somme d’un facteur dont une puissance extérieure est nulle et d’un facteur dont une puissance symétrique est nulle. Je présenterai la théorie de cette notion (purement catégorique), puis ses applications en géométrie algébrique.
LA - fre
KW - Chow groups; motives; tensor categories; parity
UR - http://eudml.org/doc/252130
ER -

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