Global rigidity of the period mapping

Let ${mathcal M}_{g,n}$ denote the moduli space of smooth, genus $ggeq 1$ curves with $ngeq 0$ marked points. Let ${mathcal A}_h$ denote the moduli space of $h$-dimensional, principally polarized abelian varieties. Let $ggeq 4$ and $hleq g$. If $F:{mathcal M}_{g,n}to{mathcal A}_h$ is a nonconstant holomorphic map then $h=g$ and $F$ is the classical period mapping, assigning to a Riemann surface $X$ its Jacobian.