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 article 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 work 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.

中文翻译：

---

代数等单变形和映射类群

一个稳定的对数连接的全等单数变形的萌芽$n$尖的属$g$曲线总是存在于解析范畴中。本文的第一部分研究在什么条件下它是代数等单数形变的解析萌化。在一些次要的技术条件下，当且仅当连接的单一性在映射类群的作用下具有有限轨道时，情况才会如此。这项工作的第二部分研究了在可约 2 级表示和属的特定情况下该动作的动力学$g>0$，允许对所有有限轨道进行分类。这两个结果都扩展了最近关于 0 类病例的结果。