On Kawai theorem for orbifold Riemann surfaces

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 Teichmuller 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 form on the $${\mathrm {PSL}}(2,\mathbb {R})$$ -character variety is a symplectic form of the Weil–Petersson metric on the Teichmuller space.