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

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 Teichmuller space for S. The total space of the holomorphic cotangent bundle of the Teichmuller 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 Teichmuller 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.