Motifs de dimension finie

André, Yves

Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Exposé no. 929, pp. 115-145.

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.

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.

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