Torelli theorem for the Deligne--Hitchin moduli space

Abstract

Fix integers g ≥ 3 and r ≥ 2, with r ≥ 3 if g = 3. Given a compact connected Riemann surface X of genus g, let M_DH(X) denote the corresponding SL(r, C) Deligne-Hitchin moduli space. We prove that the complex analytic space M_DH(X) determines (up to an isomorphism) the unordered pair {X, overline{X}} , where {overline{X}} is the Riemann surface defined by the opposite almost complex structure on X.