Comparison of Poisson structures on moduli spaces

Abstract

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