On moduli spaces of Hitchin pairs

Let X be a compact Riemann surface X of genus at-least two. Fix a holomorphic line bundle L over X. Let M be the moduli space of Hitchin pairs (E, phi is an element of H-0(End(0)(E) circle times L)) over X of rank r and fixed determinant of degree d. The following conditions are imposed: (i) deg(L) >= 2g - 2, r >= 2, and L-circle times r not equal K-X(circle times r) (ii) (r, d) = 1; and (iii) if g = 2 then r >= 6, and if g = 3 then r >= 4. We prove that that the isomorphism class of the variety M uniquely determines the isomorphism class of the Riemann surface X. Moreover, our analysis shows that M is irreducible (this result holds without the additional hypothesis on the rank for low genus).