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 Teichmüller space to the PSL (2 , 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 form on the PSL (2 , R) -character variety is a symplectic form of the Weil–Petersson metric on the Teichmüller space.