Gauge fixing for logarithmic connections over curves and the Riemann-Hilbert-Problem

1995/04/29 by Christian Gantz, Gantz, Christian, Brian Steer +1
Mathematics · #14H60 (Primary) 14H30 14F10 14F35 (Secondary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.alg-geom/9504016

openalex publication_date 1995/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We explain in detail the correspondence between algebraic connections over CP¹, logarithmic at X = x₁,...,x_n ⊂ CP¹, and flat bundles over CP¹-X with integer weighted filtrations near each x_j. Included is a gauge fixing theorem for logarithmic connections. (Thus far, one could work over any Riemann surface.) We prove a bound on the splitting type of a semi-stable logarithmic connection over CP¹. Using this we extend and simplify some results on the Riemann-Hilbert-Problem, which asks for a logarithmic connection on a holomorphically trivial bundle over CP¹, extending a given flat bundle over CP¹-X. The work is self contained and elementary, using only basic knowledge of Gauge Theory and the Birkhoff-Grothendieck-Theorem.

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