Vol. 263, No. 2, 2013

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Stable flags, trivializations and regular connections

Elie Compoint and Eduardo Corel

Vol. 263 (2013), No. 2, 283–352

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.

meromorphic connection, vector bundle, Birkhoff–Grothendieck theorem, Bruhat–Tits building, Riemann–Hilbert problem
Mathematical Subject Classification 2010
Primary: 51N30, 34M03
Secondary: 34M50
Received: 3 December 2010
Revised: 14 March 2013
Accepted: 28 March 2013
Published: 31 May 2013
Elie Compoint
Université des Sciences et Technologies - Lille 1
Cité Scientifique
59655 Villeneuve d’Ascq
Eduardo Corel
Université d’Evry-Val-d’Essonne
IBGBI, 23 boulevard de France
91037 Evry