Yves Laszlo

Conformal Blocks and Generalized Theta Functions

LetSUXr be the moduli space of rankr vector bundles with trivial determinant on a Riemann surfaceX. This space carries a natural line bundle, the determinant line bundleL. We describe a canonical isomorphism of the space of global sections ofLk with the space of conformal blocks defined in terms of representations of the Lie algebraslr(C((z))). It follows in particular that the dimension ofH0(SUXr,Lk) is given by the Verlinde formula.

Local structure of the moduli space of vector bundles over curves

We analyze the local structure of the moduli space of semi-stable bundles on a curve. In particular, a complete description of the local structure is given in the rank 2 case. We obtain as a corollary of this analysis new results about the Kummer variety and the Coble quartic associated to a canonical genus 3 curve.

On the Hitchin morphism in positive characteristic

Let X be a smooth projective curve over a field of characteristic p > 0. We show that the Hitchin morphism, which associates to a Higgs bundle its characteristic polynomial, has a nontrivial deformation over the affine line. This deformation is constructed by considering the moduli stack of t-connections on vector bundles on X and an analogue of the p-curvature, and by observing that the associated characteristic polynomial is, in a suitable sense, a p th -power.

The action of the Frobenius map on rank 2 vector bundles in characteristic 2

Let

The Frobenius map, rank 2 vector bundles and Kummer's quartic surface in characteristic 2 and 3

X

The line bundles on the moduli of parabolic G-bundles over curves and their sections

be an ordinary smooth curve defined over an algebraically closed field of characteristic 2. The absolute Frobenius induces a rational map

Un lemme de descente

F

Fibrés génériques sur le plan projectif

on the moduli space

A propos de l'espace des modules de fibrés de rang 2 sur une courbe

M_X

A non-trivial family of bundles fixed by the square of Frobenius

of rank 2 vector bundles with fixed trivial determinant. If the genus of