Catégories dérivées et géométrie birationnelle [d'après Bondal, Orlov, Bridgeland, Kawamata...]

Rouquier, Raphaël

Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Exposé no. 946, pp. 283-307.

À l’origine conçue comme un outil technique, la catégorie dérivée des faisceaux cohérents d’une variété algébrique est apparue lors de ces dix dernières années comme un invariant important dans l’étude birationnelle des variétés algébriques. Des problèmes d’invariance birationnelle et de minimisation de la catégorie dérivée sont apparus, inspirés par la conjecture homologique de symétrie miroir de Kontsevich et le programme de Mori de modèles minimaux pour les variétés algébriques. Nous présenterons les conjectures générales et leur preuve en dimension $3$ et pour des flops particuliers.

Originally a technical tool, the derived category of coherent sheaves over an algebraic variety has become over the last twenty years an important invariant in the birational study of algebraic varieties. Problems of birational invariance and of minimization of the derived category have appeared, inspired by Kontsevich's homological mirror symmetry conjecture and Mori's minimal model program. We present the main conjectures and their proofs in dimension 3 and for particular classes of flops.

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