Monodromy of Schwarzian equations with regular singularities

Faraco, Gianluca; Gupta, Subhojoy

doi:10.2140/gt.2025.29.549

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Monodromy of Schwarzian equations with regular singularities

Gianluca Faraco and Subhojoy Gupta

Geometry & Topology 29 (2025) 549–618

DOI: 10.2140/gt.2025.29.549

Abstract

Let $S$ be a punctured surface of finite type and negative Euler characteristic. We determine all possible representations $ρ : π_{1} (S) \to {PSL}_{2} (ℂ)$ 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.

Keywords

Schwarzian equations, complex projective structures

Mathematical Subject Classification

Primary: 57M50

References

Publication

Received: 4 August 2022

Revised: 5 June 2023

Accepted: 23 September 2023

Published: 12 March 2025

Proposed: Anna Wienhard

Seconded: Benson Farb, David Fisher

Authors

Gianluca Faraco
Dipartimento di Matematica e Applicazioni U5 Università degli Studi di Milano-Bicocca Milan Italy

Subhojoy Gupta
Department of Mathematics Indian Institute of Science Bangalore India

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Volume 29, issue 2 (2025)