Picard theorems for moduli spaces of polarized varieties

As a result of our study of the hyperbolicity of the moduli space of polarized manifold, we give a general big Picard theorem for a holomorphic curve on a log-smooth pair $(X,D)$ such that $W=X\setminus D$ admits a Finsler pseudometric that is strongly negatively curved when pulled back to the curve. We show, by some refinements of the classical Viehweg-Zuo construction, that this latter condition holds for the base space $W$, if nonsingular, of any algebraic family of polarized complex projective manifolds with semi-ample canonical bundles whose induced moduli map $\phi$ to the moduli space of such manifolds is generically finite and any $\phi$-horizontal holomorphic curve in $W$. This yields the big Picard theorem for any holomorphic curves in the base space $U$ of such an algebraic family by allowing this base space to be singular but with generically finite moduli map. An immediate and useful corollary is that any holomorphic map from an algebraic variety to such a base space $U$ must be algebraic, i.e., the corresponding holomorphic family must be algebraic. We also show the related algebraic hyperbolicity property of such a base space $U$, which generalizes previous Arakelov inequalities and weak boundedness results for moduli stacks and offers, in addition to the Picard theorem above, another evidence in favor of the hyperbolic embeddability of such an $U$.