Le Potier  Lectures on vector bundles

Название: 
Lectures on vector bundles 
Автор: 
Le Potier 
Категория: 
Математика

Тип: 
Книга 
Дата: 
30.12.2008 23:44:33 
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Описание: 
This work consists of two courses treating the moduli spaces of vector bundles. The first part is an elementafy DEA course which we have given at the Universite Denis Diderot (Paris VII) in the Spring of 1991: the subject tackled is the classification of vector bundles on algebraic curves, the construction of the moduli spaces of stable bundles and the first properties of these moduli spaces. In particular, one will find the smoothness criteria and the existence conditions of stable bundles of given rank and degree. It consists of some reminders of facts from algebraic geometry and algebraic topology, the construction, in the case of curves, of the HilbertGrothendieck scheme of coherent quotient sheaves of a given bundle, and to treat this we also give a succinct description of Mumford's geometric invariant theory.
The second part is a more advanced course given at the school "Vector Bundles on Surfaces" organized by EUROPROJ (European projective geometry network) and CIMI (Centre International pour les Mathematiques et l'lnformatique) at Nice SophiaAntipolis (1418 June 1993). The subject we treated here centred on the structure of the moduli space of semistable sheaves on the projective plane. In particular, we establish the existence conditions of semistable sheaves of given rank and Chern classes and deal with the question of irreducibility and describe the Picard group of the moduli spaces. The construction of the moduli space of semistable sheaves is sketched in the more general situation of projective algebraic surfaces: it is fairly similar to the construction we shall see in the first part for the case of curves once we have established the fact that the family of semistable sheaves with fixed Hilbert polynomial is bounded; in fact, it generalizes without much dificulty to higher dimensional varieties. The fact that this family is bounded rests on the theorems dealing with restriction of semistable sheaves to curves which 
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