Monodromy of Schwarzian equations with regular singularities

Abstract

Let S be a punctured surface of finite type and negative Euler characteristic. We determine all possible representations ρ : π 1 (S ) → PSL 2 (C) that arise as the monodromy of the Schwarzian equation on S with regular singularities at the punctures. Equivalently, we determine the holonomy representations of complex projective structures on S , whose Schwarzian derivatives (with respect to some uniformizing structure) have poles of order at most two at the punctures. Following earlier work that dealt with the case when there are no apparent singularities, our proof reduces to the case of realizing a degenerate representation with apparent singularities. This mainly involves explicit constructions of complex affine structures on punctured surfaces, with prescribed holonomy. As a corollary, we determine the representations that arise as the holonomy of spherical metrics on S with cone-points at the punctures.

Spherical surfaces with conical points: systole inequality and moduli spaces with many connected components

“…On the other hand, if some ϑ i is integer, then [ρ] sends every β ′ i ∈ B i to ±I and so ρ(β ′ i ) does not determine an axis of rotation. In order to add the extra piece of information consisting of axes of rotation of ρ(β) for all β ∈ B, we introduce decorations (a similar construction can be found in [22], [19], [11], [1], [10]). We call such couple (ρ, A) a decorated homomorphism.…”

“…However such result only partially answers Question 1.19 for two reasons. First, Corollary D of [10] only deals with non-decorated representations. Moreover, the angles of the conical points of the surface whose monodromy realizes the given representation are only determined up to adding multiples of 2π.…”