Algebraic theory of formal regular-singular connections with parameters

Abstract

This paper is divided into two parts. The first is a review, through categorical lenses, of the classical theory of regular-singular differential systems over $C((x))$ and ${\mathbb{P}}_C^1\setminus \{0,\infty \}$, where $C$ is algebraically closed and of characteristic zero. It aims to read the existing classification results as an equivalence between regular-singular systems and representations of the group $\mathbb{Z}$. In the second part, we deal with regular-singular connections over $R((x))$ and ${\mathbb{P}}_R^1\setminus \{0,\infty \}$, where $R=C[[t_1,\dots ,t_r]]/I$. The picture we offer shows that regular-singular connections are equivalent to representations of $\mathbb{Z}$, now over $R$.