On Kawai theorem for orbifold Riemann surfaces

TL;DR

This paper generalizes Kawai's theorem to orbifold Riemann surfaces, providing explicit formulas for symplectic forms and extending Goldman's theorem to orbifold cases, connecting character varieties and Weil-Petersson metrics.

Contribution

It introduces a new formula for the differential of a holomorphic map in the orbifold setting, enabling explicit evaluation of symplectic forms and extending classical theorems.

Findings

Generalization of Kawai's theorem to orbifold Riemann surfaces.

Explicit computation of the pullback of Goldman symplectic form.

Extension of Goldman's theorem relating character varieties and Weil-Petersson metric.

Abstract

We prove a generalization of Kawai theorem for the case of orbifold Riemann surface. The computation is based on a formula for the differential of a holomorphic map from the cotangent bundle of the Teichm\"uller space to the $\mathrm{PSL}(2,\mathbb{C})$-character variety, which allows to evaluate explicitly the pullback of Goldman symplectic form in the spirit of Riemann bilinear relations. As a corollary, we obtain a generalization of Goldman's theorem that the pullback of Goldman symplectic from on $\mathrm{PSL}(2,\mathbb{R})$-character variety is a symplectic form of the Weil-Petersson metric on the Teichm\"uller space.