  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 $U$.
