Solutions alg\'ebriques. Solutions alg\'ebriques partielles des \'equations isomonodromiques sur les courbes de genre $2$

TL;DR

This paper explores the construction of partial algebraic solutions to isomonodromic equations on genus 2 curves, adapting methods involving Hurwitz families to classify cases with Zariski dense monodromy.

Contribution

It introduces a new approach to find algebraic solutions of isomonodromic equations on genus 2 curves using Hurwitz families, extending previous methods.

Findings

Classified all cases with Zariski dense monodromy.

Developed an adapted method based on Andreev and Kitaev's approach.

Identified conditions for partial algebraic solutions.

Abstract

On \'etudie la possibilit\'e de construire des solutions alg\'ebriques partielles des \'equations d'isomonodromie pour les connections holomorphes de rang $2$ sur les courbes de genre $2$ en adaptant la m\'ethode d'Andreev et Kitaev par les familles de Hurwitz. Nous classifions tous les cas o\`u la connection est \`a monodromie Zariski dense.