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

TL;DR

This paper demonstrates a biholomorphic correspondence between the moduli space of orbifold projective structures and the holomorphic cotangent bundle of Teichmüller space, preserving their natural symplectic forms.

Contribution

It establishes a symplectic-compatible biholomorphic mapping between the moduli space and the cotangent bundle using uniformization techniques.

Findings

The moduli space is biholomorphic to the cotangent bundle of Teichmüller space.

The symplectic structures on both spaces are compatible under this mapping.

Similar results hold for Schottky and Earle uniformizations.

Abstract

Let S be a compact connected oriented orbifold surface We show that using Bers simultaneous uniformization, the moduli space of projective structure on S can be mapped biholomorphically onto the total space of the holomorphic cotangent bundle of the Teichm\"uller space for S. The total space of the holomorphic cotangent bundle of the Teichm\"uller space is equipped with the Liouville symplectic form, and the moduli space of projective structures also has a natural holomorphic symplectic form. The above identification 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.