Holonomy of complex projective structures on surfaces with prescribed branch data

TL;DR

This paper characterizes the holonomy representations of branched complex projective structures on closed surfaces with fixed branch data, including spherical and affine structures with prescribed conical angles.

Contribution

It provides a complete description of which representations arise as holonomies for these structures with specified branch divisors.

Findings

Computed holonomies of spherical metrics with prescribed conical angles.

Determined holonomies of affine structures with fixed conical angles.

Characterized the representations of the fundamental group corresponding to these structures.

Abstract

We characterize the representations of the fundamental group of a closed surface to $\mathrm{PSL}_2(\mathbb C)$ that arise as the holonomy of a branched complex projective structure with fixed branch divisor. In particular, we compute the holonomies of the spherical metrics with prescribed integral conical angles and the holonomies of affine structures with fixed conical angles on closed surfaces.