Global rigidity of the period mapping

Let M g , n {\mathcal M}_{g,n} denote the moduli space of smooth, genus g ≥ 1 g\geq 1 curves with n ≥ 0 n\geq 0 marked points. Let A h {\mathcal A}_h denote the moduli space of h h -dimensional, principally polarized abelian varieties. Let g ≥ 3 g\geq 3 and h ≤ g h\leq g . If F : M g , n → A H F:{\mathcal M}_{g,n} \to {\mathcal A}_H is a nonholomorphic map, then h = g h=g and F F is the classical period mapping, assigning to a Riemann surface X X its Jacobian.