Stable flags, trivializations and regular connections

Elie Compoint and Eduardo Corel

Vol. 263 (2013), No. 2, 283–352

DOI: 10.2140/pjm.2013.263.283

Abstract

We study stalkwise modifications of a holomorphic vector bundle endowed with a meromorphic connection on a compact Riemann surface. We introduce the notion of Birkhoff–Grothendieck trivialization, in the case of the Riemann sphere, and show that its computation corresponds to shortest paths in some local affine Bruhat–Tits building. We use this to compute how the type of a bundle changes under stalk modifications, and give several corresponding algorithmic procedures. We finally deduce from these results some applications to the Riemann–Hilbert problem.

Keywords

meromorphic connection, vector bundle, Birkhoff–Grothendieck theorem, Bruhat–Tits building, Riemann–Hilbert problem

Mathematical Subject Classification 2010

Primary: 51N30, 34M03

Secondary: 34M50

Milestones

Received: 3 December 2010

Revised: 14 March 2013

Accepted: 28 March 2013

Published: 31 May 2013