Meromorphic Projective Structures: Signed Spaces, Grafting and Monodromy

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.