connections on stable bundles
by
Olivier Mathieu
Universite de Lyon

Let X be a Riemmann surface, i.e a genus g surface with a complex structure.

DEFINITION: An holomorphic vector bundle E is called stable if c(E)=0 but c(F)<0 for any proper subbundle F.

Here ßtable" stands for stable of slope 0. The notation c(F) is the first Chern class of F. The meaning of "c(F)<0" is explained by the natural identification of the second cohomology group of X with Z.

Narashiman and Seshadri have shown that a stable bundle admits a unique hermitian holomorphic connection. When X and E come from an algebraic curve and an holomorphic bundle defined over a number field K, the previous theorem attach to each infinite place of K a certain connection.

We show a similar statement for finite places of K. Then we state a conjecture about the algebraicity of solutions of certain differential equations.

Date received: July 8, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cals-17.