Fuchsian moduli on a Riemann surface - its poisson structure and Poincaré-Lefschetz duality

Abstract

The moduli space of Fuchsian projective connections on a closed Riemann surface admits a Poisson structure. The moduli space of projective monodromy representations on the punctured Riemann surface also admits a Poisson structure which arises from the Poincaré-Lefschetz duality for cohomology. We shall show that the former Poisson structure coincides with the pull-hack of the latter by the projective monodromy map. This result explains intrinsically why a Hamiltonian structure arises in the monodromy preserving deformation. © 1992 by Pacific Journal of Mathematics.