Stable Flags and the Riemann-Hilbert Problem

TL;DR

This paper investigates the Riemann-Hilbert problem on the Riemann sphere, establishing a bijection with stable local filtrations, introducing Birkhoff-Grothendieck trivialisation, and providing algorithms for stalk modifications.

Contribution

It introduces the notion of Birkhoff-Grothendieck trivialisation and links it to geodesic paths in affine Bruhat-Tits buildings, offering new computational methods.

Findings

Solutions correspond to stable local filtrations under monodromy

Birkhoff-Grothendieck trivialisation relates to geodesic paths in affine buildings

Algorithms for stalk modifications of vector bundles

Abstract

We tackle the Riemann-Hilbert problem on the Riemann sphere as stalk-wise logarithmic modifications of the classical R\"ohrl-Deligne vector bundle. We show that the solutions of the Riemann-Hilbert problem are in bijection with some families of local filtrations which are stable under the prescribed monodromy maps. We introduce the notion of Birkhoff-Grothendieck trivialisation, and show that its computation corresponds to geodesic 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.