Meromorphic connections on the projective line with specified local behavior

TL;DR

This paper explores how specified local behaviors at singular points influence the existence and uniqueness of global meromorphic connections on the complex projective line, addressing various types of singularities and their moduli spaces.

Contribution

It provides a comprehensive discussion on the conditions under which prescribed local data determine global meromorphic connections, extending to irregular singularities and rigidity problems.

Findings

Conditions for existence of connections with given local types

Criteria for uniqueness and rigidity of connections

Extensions to irregular singularities and complex singularity structures

Abstract

A meromorphic connection on the complex projective line induces formal connections at each singular point, and these formal connections constitute the local behavior at the singularities. In this primarily expository paper, we discuss the extent to which specified local behavior at singular points determines the global connection. In particular, given a finite set of points and a collection of ``formal types'' at these points, does there exist a moduli space of meromorphic connections with this local behavior, and if so, when is this moduli space nonempty or a singleton? In this paper, we discuss variants of these problems (for example, the Deligne--Simpson and rigidity problems) as the allowed singularities get progressively more complicated: first connections with only regular singularities, next connections with additional unramified irregular singularities allowed, and finally the general case.