Birational invariants of algebraic varieties

We extend the notion of Iitaka dimension of a line bundle, and deene a family of dimensions indexed by weights, for vector bundles of arbitrary ranks on complex varieties. In the case of the cotangent bundle, we get a family of birational invariants, including the Kodaira dimension and the cotangent genus introduced by Sakai. We treat in detail several classes of examples, such as complete intersections, low codi-mensional subvarieties of projective spaces or tori, and projective bundles. We nally examine to what extent our invariants allow to reene the Kodaira-Enriques birational classiication of surfaces.