Let X and X' be a smooth projective curves over the complex numbers, the classical Torelli theorem says that the Jacobian J(X) together with the polarization given by the theta divisor determines the
curve X, i.e. if J(X) and J(X') are isomorphic as polarized abelian varieties then X and X' are also isomorphic.

A similar result holds for the moduli spaces of parabolic bundles (V. Balaji, I. Biswas, S. del Baño). Given a finite subset S of X, and fixing a numerical data, known as parabolic structure, denote M (resp.
M') the moduli space of parabolic semi-stable bundles with that fixed parabolic structure on S in X , then M and M' are isomorphic if and only if there is an isomorphism between X and X' which takes S to S'. We will deal with the same problem for parabolic Higgs bundles, i.e.Higgs bundles with a parabolic structure. This is work in progress with T. Gómez.