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 H0(X,ωX−1(−D)) ≠ 0. Let us denote by ?ℬ 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 ?ℬ is a non-singular quasi-projective variety naturally endowed with a family of holomorphic Poisson structures parametrized by the global sections of ωX−1(−D). This result is the natural generalization to the moduli spaces of parabolic vector bundles of the results obtained in [B2] 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 ?ℬ.