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Birational invariants of algebraic varieties. (English) Zbl 0811.14008
We extend the notion of Iitaka dimension of a line bundle, and define 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 codimensional subvarieties of projective spaces or tori, and projective bundles.
We finally examine to what extent our invariants allow the refinement of the Kodaira-Enriques birational classification of surfaces.
Reviewer: L.Manivel

14E05 Rational and birational maps
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J10 Families, moduli, classification: algebraic theory
14M07 Low codimension problems in algebraic geometry
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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