. A G ] 1 3 Fe b 20 07 UNITARY LOCAL SYSTEMS , MULTIPLIER IDEALS , AND POLYNOMIAL PERIODICITY OF HODGE NUMBERS

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 h q,0 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. We derive a geometric characterization of finite abelian covers, which recovers the ones of Namba and Pardini. We use this, for example, to prove a conjecture of Ligbober about Hodge numbers of abelian covers.