Comparison of Poisson structures on moduli spaces

Let X be a complex irreducible smooth projective curve, and let L be an algebraic line bundle on X with a nonzero section σ_0. Let M denote the moduli space of stable Hitchin pairs (E, θ), where E is an algebraic vector bundle on X of fixed rank r and degree δ, and θ ∈ H^0(X, End(E) ⊗ K_X ⊗ L). Associating to every stable Hitchin pair its spectral data, an isomorphism of M with a moduli space P of stable sheaves of pure dimension one on the total space of K_X ⊗ L is obtained. Both the moduli spaces P and M are equipped with algebraic Poisson structures, which are constructed using σ_0 . Here we prove that the above isomorphism between P and M preserves the Poisson structures.