The monodromy of meromorphic projective structures

TL;DR

This paper investigates the monodromy map of meromorphic projective structures with poles on surfaces, showing its image lies within cluster chart domains and exploring related open questions.

Contribution

It introduces a monodromy map for meromorphic projective structures and analyzes its image within the framework of cluster charts.

Findings

Monodromy map's image is contained in cluster chart domains

Establishes a parameterization of structures via local systems

Discusses open questions about the monodromy map's properties

Abstract

We study projective structures on a surface having poles of prescribed orders. We obtain a monodromy map from a complex manifold parameterising such structures to the stack of framed $\mathrm{PGL}_2(\mathbb{C})$ local systems on the associated marked bordered surface. We prove that the image of this map is contained in the union of the domains of the cluster charts. We discuss a number of open questions concerning this monodromy map.