Niveau: Supérieur, Doctorat, Bac+8
Complex Algebraic Geometry MSRI Publications Volume 28, 1995 Vector Bundles on Curves and Generalized Theta Functions: Recent Results and Open Problems ARNAUD BEAUVILLE Abstract. The moduli spaces of vector bundles on a compact Riemann surface carry a natural line bundle, the determinant bundle. The sections of this line bundle and its multiples constitute a non-abelian generalization of the classical theta functions. New ideas coming from mathematical physics have shed a new light on these spaces of sections—allowing notably to compute their dimension (Verlinde's formula). This survey paper is devoted to giving an overview of these ideas and of the most important recent results on the subject. Introduction It has been known essentially since Riemann that one can associate to any compact Riemann surface X an abelian variety, the Jacobian JX , together with a divisor (well-dened up to translation) that can be dened both in a geometric way and as the zero locus of an explicit function, the Riemann theta function. The geometry of the pair (JX; ) is intricately (and beautifully) related to the geometry of X . The idea that higher-rank vector bundles should provide a non-abelian ana- logue of the Jacobian appears already in the influential paper [We] of A. Weil (though the notion of vector bundle does not appear as such in that paper!).
- vector bundles should
- vector bundles
- generalized theta
- jacobian jx
- section has
- following diagram commutative
- complex algebraic
- geometry msri
- theta divisor
- functions