Journal of Symplectic Geometry

Volume 15 (2017)

Number 3

On the symplectic structure over a moduli space of orbifold projective structures

Pages: 621 – 643

DOI: https://dx.doi.org/10.4310/JSG.2017.v15.n3.a1

Authors

Pablo Arés-Gastesi (Department of Applied Mathematics and Statistics, School of Economics and Business, Universidad CEU San Pablo, Madrid, Spain)

Indranil Biswas (School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India)

Abstract

Let $S$ be a compact connected oriented smooth orbifold surface. We show that using Bers simultaneous uniformization, the moduli space of projective structures on $S$ can be mapped biholomorphically onto the total space of the holomorphic cotangent bundle of the Teichmüller space for $S$. The total space of the holomorphic cotangent bundle of the Teichmüller space is equipped with the Liouville holomorphic symplectic form, and the moduli space of projective structures also has a natural holomorphic symplectic form. The above identification between the moduli space of projective structures on $S$ and the holomorphic cotangent bundle of the Teichmüller space for $S$ is proved to be compatible with these symplectic structures. Similar results are obtained for biholomorphisms constructed using uniformizations provided by Schottky groups and Earle’s version of simultaneous uniformization.

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The second-named author is partially supported by a J. C. Bose Fellowship.

Received 18 December 2013

Published 8 September 2017