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

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 Poincare-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. Introduction. It has been known that a Hamiltonian structure arises in the theory of monodromy preserving deformation of meromorphic differential equations. See [KO], [O]. However it has not yet been known why such a Hamiltonian structure does arise. Our result in the present paper will explain the reason clearly and intrinsically. Rather it will be even self-evident from our point of view why such a Hamiltonian structure arises.