Algebraic isomonodromic deformations and the mapping class group

TL;DR

This paper explores conditions under which analytic isomonodromic deformations are algebraic, linking this to the finiteness of monodromy orbits under the mapping class group, and classifies finite orbits for certain reducible representations.

Contribution

It establishes a criterion connecting algebraic isomonodromic deformations with finite monodromy orbits and classifies these orbits for reducible rank 2 cases in higher genus.

Findings

Algebraic isomonodromic deformations occur iff monodromy has finite mapping class group orbit.

Classified all finite orbits for reducible rank 2 representations in genus g > 0.

Extended genus 0 results to higher genus cases.

Abstract

The germ of the universal isomonodromic deformation of a logarithmic connection on a stable n-pointed genus g curve always exists in the analytic category. The first part of this paper investigates under which conditions it is the analytic germification of an algebraic isomonodromic deformation. Up to some minor technical conditions, this turns out to be the case if and only if the monodromy of the connection has finite orbit under the action of the mapping class group. The second part of this paper studies the dynamics of this action in the particular case of reducible rank 2 representations and genus g > 0, allowing to classify all finite orbits. Both of these results extend recent ones concerning the genus 0 case.