Meromorphic projective structures: signed spaces, grafting and monodromy

Ghosh, Spandan; Gupta, Subhojoy

doi:10.2140/agt.2025.25.4787

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Meromorphic projective structures: signed spaces, grafting and monodromy

Spandan Ghosh and Subhojoy Gupta

Algebraic & Geometric Topology 25 (2025) 4787–4826

DOI: 10.2140/agt.2025.25.4787

Abstract

A meromorphic quadratic differential on a compact Riemann surface defines a complex projective structure away from the poles via the Schwarzian equation. In this article we first prove the analogue of Thurston’s grafting theorem for the space of such structures with signings at regular singularities. This extends previous work of Gupta–Mj which only considered irregular singularities. We also define a framed monodromy map from the signed space extending work of Allegretti–Bridgeland, and we characterize the ${PSL}_{2} (ℂ)$ -representations that arise as holonomy, generalizing results of Gupta–Mj and Faraco–Gupta. As an application of our grafting theorem, we also show that the monodromy map to the moduli space of framed representations (as introduced by Fock–Goncharov) is a local biholomorphism, proving a conjectured analogue of a result of Hejhal.

Keywords

complex projective structures, meromorphic quadratic differentials, signed measured laminations

Mathematical Subject Classification

Primary: 30F30, 57M50

References

Publication

Received: 4 December 2023

Revised: 30 November 2024

Accepted: 31 December 2024

Published: 20 November 2025

Authors

Spandan Ghosh
The Mathematical Institute Oxford University Oxford United Kingdom

Subhojoy Gupta
Department of Mathematics Indian Institute of Science Bangalore India

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Volume 25, issue 8 (2025)