Global rigidity of the period mapping

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