| Summary: | In this paper, we study various hyperbolicity properties for a quasi-compact
K\"ahler manifold $U$ which admits a complex polarized variation of Hodge
structures so that each fiber of the period map is zero-dimensional. In the
first part, we prove that $U$ is algebraically hyperbolic and that the
generalized big Picard theorem holds for $U$. In the second part, we prove that
there is a finite \'etale cover $\tilde{U}$ of $U$ from a quasi-projective
manifold $\tilde{U}$ such that any projective compactification $X$ of
$\tilde{U}$ is Picard hyperbolic modulo the boundary $X-\tilde{U}$, and any
irreducible subvariety of $X$ not contained in $X-\tilde{U}$ is of general
type. This result coarsely incorporates previous works by Nadel, Rousseau,
Brunebarbe and Cadorel on the hyperbolicity of compactifications of quotients
of bounded symmetric domains by torsion-free lattices.
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