Cursos de Verão em Geometria 

23/06/2010, 15:30 — 17:00 — Sala P3.10, Pavilhão de Matemática
Vicente Muñoz, Universidad Complutense de Madrid

Moduli spaces of pairs: Torelli theorem

We shall study moduli spaces of vector bundles over a complex curve, and moduli spaces of pairs formed by a vector bundle and a global section. There is a concept of stability for pairs which depends on a real parameter. These moduli spaces suffer a birational transformation when the parameter crosses certain critical values. We shall see how properties of the algebraic structure of the mentioned moduli spaces (such as Hodge numbers, Hodge structures, Torelli theorems) can be deduced from a careful study of these birational transformations.

We use an inductive argument on the rank involving the moduli spaces of bundles and the moduli of pairs to get topological and geometrical properties like: irreducibility, Picard groups, or "Torelli theorems" which say that the moduli space determines the curve.