Leon Takhtajan (Stony Brook), Symplectic geometry of the space of complex projective structures on orbifold Riemann surfaces

The space of complex projective connections on Riemann surfaces is an affine bundle, modeled on the holomorphic tangent bundle of the moduli space, and carries a holomorphic symplectic form - Liouville (2,0) form. The monodromy map is a holomorphic mapping of this space into the PSL(2,C)-character variety of an orbifold Riemann surface. The latter variety carries a natural symplectic form, introduced by Bill Goldman for compact for the compact surface. An old theorem (Kawai, 1996) states that in the compact case a pullback of Goldman symplectic form is the Liouville (2,0) form. However, Kawai’s prove was not very insightful. I will explain how one can determine the differential of the monodromy map and to compute the pullback directly, in the spirit of Riemann bilinear relations. This elucidates the symplectic structure of these varieties and proves a generalization of Kawai theorem - a pullback of Goldman symplectic form is the Liouville (2,0) form - for orbifold Riemann surfaces.