On linear differential equations with reductive Galois group

TL;DR

This paper classifies certain linear differential equations with reductive Galois groups on Riemann surfaces by associating projective varieties and extends Klein's classical results to a broader class.

Contribution

It introduces a classification framework for connections with reductive Galois groups using associated projective varieties and ruled surfaces.

Findings

Connections with curve-associated projective varieties are classified up to projective equivalence.

Such connections are pullbacks of a Standard connection.

Extends Klein's classification from second-order equations to broader classes.

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.