Symplectic geometry of the moduli space of projective structures in homological coordinates

Abstract

We study the symplectic geometry of the space of linear differential equations with holomorphic coefficients of the form $$\varphi ''-u\varphi =0$$ on Riemann surfaces of genus g. This space coincides with the moduli space of projective connections which is an affine bundle modelled on the cotangent bundle $$T^*{\mathcal {M}}_g$$ . We show that for several choices of the origin, or base, holomorphic projective connection (such as Bergman, Wirtinger or Schottky) the canonical Poisson structure on $$T^*{\mathcal {M}}_g$$ induces the Goldman bracket on the monodromy character variety. These different choices give rise to equivalent symplectic structures on the space of projective connections but different symplectic polarizations; we find the corresponding generating functions. Combined with a prior theorem of Kawai, our results show the symplectic equivalence between the embeddings of $$T^*{\mathcal {M}}_g$$ induced by the Bers and Bergman projective connections into the space of projective structures. The main technical tools are variational formulas with respect to homological Darboux coordinates on the space of holomorphic quadratic differentials. In particular, we get a new system of differential equations for the Prym matrix of the canonical two-sheeted covering of a Riemann surface defined by a quadratic differential.