Holonomy of complex projective structures on surfaces with prescribed branch data

Abstract

We characterize the representations of the fundamental group of a closed surface to ${\mathrm{PSL}}_2\left(C\right)$ 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.