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

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

1995-04-29

arxiv.org/abs/alg-geom/9504016v1

Mathematics

Algebraic Geometry

29 pages, Latex 2.09

Scientific paper

We explain in detail the correspondence between algebraic connections over CP^{1}, logarithmic at X = { x_{1},...,x_{n} } \subset CP^{1}, and flat bundles over CP^{1}-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 CP^{1}. 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 CP^{1}, extending a given flat bundle over CP^{1}-X. The work is self contained and elementary, using only basic knowledge of Gauge Theory and the Birkhoff-Grothendieck-Theorem.

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