On Kawai theorem for orbifold Riemann surfaces

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 $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{C})$-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 $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$-character variety is a symplectic form of the Weil-Petersson metric on the Teichm"uller space.