Poisson Structures on Moduli Spaces of Parabolic Bundles on Surfaces

Let X be a smooth complex projective surface and D an effective divisor on X such that H^0(X,\omega_X^{-1}(-D)) \ne 0. Let us denote by PB the moduli space of stable parabolic vector bundles on X with parabolic structure over the divisor D (with fixed weights and Hilbert polynomials). We prove that the moduli space PB is a non-singular quasi-projective variety naturally endowed with a family of holomorphic Poisson structures parametrized by the global sections of \omega_X^{-1}(-D). This result is the natural generalization to the moduli spaces of parabolic vector bundles of the results obtained by the author for the moduli spaces of stable sheaves on a Poisson surface. We also give, in some special cases, a detailed description of the symplectic leaf foliation of the Poisson manifold PB.