Comparison of Poisson structures on moduli spaces

Let X be a complex irreducible smooth projective curve, and let $${{\mathbb {L}}}$$ L be an algebraic line bundle on X with a nonzero section $$\sigma _0$$ σ 0 . Let $${\mathcal {M}}$$ M denote the moduli space of stable Hitchin pairs $$(E,\, \theta )$$ ( E , θ ) , where E is an algebraic vector bundle on X of fixed rank r and degree $$\delta $$ δ , and $$\theta \, \in \, H^0(X,\, {\mathcal {E}nd}(E)\otimes K_X\otimes {{\mathbb {L}}})$$ θ ∈ H 0 ( X , E n d ( E ) ⊗ K X ⊗ L ) . Associating to every stable Hitchin pair its spectral data, an isomorphism of $${\mathcal {M}}$$ M with a moduli space $${\mathcal {P}}$$ P of stable sheaves of pure dimension one on the total space of $$K_X\otimes {{\mathbb {L}}}$$ K X ⊗ L is obtained. Both the moduli spaces $${\mathcal {P}}$$ P and $${\mathcal {M}}$$ M are equipped with algebraic Poisson structures, which are constructed using $$\sigma _0$$ σ 0 . Here we prove that the above isomorphism between $${\mathcal {P}}$$ P and $${\mathcal {M}}$$ M preserves the Poisson structures.