|Date||May 15, 2018|
The monodromy map sends a projective structure to a representation of fundamental group of the Riemann surface into PSL(2,C) defined up to overall conjugation. In other words we have a map from T^*M_g to the SL(2,C)-character variety which depends on the origin section chosen for the moduli space of projective structures. We will discuss previous work regarding this monodromy map and present joint work with Bertola and Korotkin which proves the map is a symplectomorphism with base Bergman, Schottky, or Wirtinger projective connection when the character variety is equipped with the Goldman bracket. Comparing our results with Kawai 96 (and more recently Loustau 15, Takhtajan 17) we propose a generating function (describing the change of Darboux coordinates) for the equivalence between the symplectic structures induced from the base Bergman projective connection versus the base Bers projective connection. We hope to discuss some open questions resulting from this work.