  The space of unitary local systems of rank one on the complement of an
arbitrary divisor in a complex projective algebraic variety can be described in
terms of parabolic line bundles. We show that multiplier ideals provide natural
stratifications of this space. We prove a structure theorem for these
stratifications in terms of complex tori and convex rational polytopes,
generalizing to the quasi-projective case results of Green-Lazarsfeld and
Simpson. As an application we show the polynomial periodicity of Hodge numbers
of congruence covers in any dimension, generalizing results of E. Hironaka and
Sakuma. We extend the structure theorem and polynomial periodicity to the
setting of cohomology of unitary local systems. In particular, we obtain a
generalization of the polynomial periodicity of Betti numbers of unbranched
congruence covers due to Sarnak-Adams. We derive a geometric characterization
of finite abelian covers, which recovers the classic one and the one of
Pardini. We use this, for example, to prove a conjecture of Libgober about
Hodge numbers of abelian covers.
