On linear differential equations with reductive Galois group

Abstract

Given a connection on a meromorphic vector bundle over a compact Riemann surface with reductive Galois group, we associate to it a projective variety. Connections such that their associated projective variety are curves can be classified, up to projective equivalence, using ruled surfaces. In particular, such a meromorphic connection is the pullbacks of a Standard connection. This extend a similar result by Klein for second-order ordinary linear differential equations to a broader class of equations.