In the paper we study various hyperbolicity properties for the 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 unramified cover U~ of U from a quasi-projective
manifold U~ so that any projective compactification X of U~
is Picard hyperbolic modulo the boundary XβU~, and any irreducible
subvariety of X not contained in XβU~ 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.Comment: 30 pages. V3, main results are improved: no monodromy assumptions are
needed. Comments are very welcome