Projective Isomonodromy and Galois Groups

TL;DR

This paper introduces the concept of projective isomonodromy as a special monodromy deformation in linear differential equations, providing algebraic criteria linked to Galois groups, exemplified through the Darboux-Halphen equation.

Contribution

It defines projective isomonodromy, establishes algebraic conditions for it, and connects it to Galois groups, expanding understanding of monodromy deformations.

Findings

Defined projective isomonodromy using Darboux-Halphen equation

Provided algebraic criteria based on Picard-Vessiot groups

Linked monodromy deformations to Galois group structures

Abstract

In this article we introduce the notion of projective isomonodromy, which is a special type of monodromy evolving deformation of linear differential equations, based on the example of the Darboux-Halphen equation. We give an algebraic condition for a paramaterized linear differential equation to be projectively isomonodromic, in terms of the derived group of its parameterized Picard-Vessiot group.