Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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G. Faraco - S. Gupta

Monodromy of Schwarzian equations with regular singularities

created by faraco on 09 Nov 2022
modified on 22 Jan 2024


Accepted Paper

Inserted: 9 nov 2022
Last Updated: 22 jan 2024

Journal: Geometry and Topology
Year: 2023

ArXiv: 2109.04044 PDF


Let $S$ be a punctured surface of finite type and negative Euler characteristic. We determine all possible representations $\rho:\pi_1(S) \to \text{PSL}_2(\mathbb{C})$ 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.

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