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 PSL 2 (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.

Monodromy of Schwarzian equations with regular singularities

“…An alternative proof. Proposition 5.4 is also a consequence of the recent work of Le Fils in [LF21], as we shall now describe. Indeed, that paper provides necessary and sufficient conditions for a representation ρ : π 1 (S g ) −→ PSL 2 (C) to appear as the monodromy of some branched projective structure with prescribed singularities; here we are interested in the case of a single branch-point.…”

“…Remark. Even in this case there is an alternative proof can be derived from the results of [LF21]. As in the preceding discussion, regard the representation ρ : π 1 (S g, 1 ) −→ PSL 2 (C) with trivial monodromy around the puncture as a representation ρ : π 1 (S g ) −→ PSL 2 (C).…”

“…Our proof shall deal with the co-axial case and non co-axial case separately. In the final subsection we also provide an alternative proof using Le Fils' results from [LF21]. Before moving to the proof of Proposition 5.4, we shall need some technical results.…”

“…The sixth and last obstruction listed in [LF21] concerns g = 2 and dihedral (but not affine) representations, and rules out the possibility of a single branch-point of order two. We shall consider dihedral representations in the next section §6.…”

“…In these cases, even when the monodromy is affine, we obtain a projective structure with the desired monodromy, which is not necessarily affine. Indeed, the recent work of [LF21] shows that in the case of a once-punctured surface, there are some further exceptions to the existence of an affine structure (see [LF21, Proposition 1.5]). In fact, this case can also be handled using the main result of [LF21]; we describe this alternative approach in §5.4.4, which involves proving the existence of a projective structure with the desired monodromy and a single branch-point, except when the monodromy group is of order two (c.f.…”