Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers

Abstract

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.

final version, to appear in Adv. Math

References in corpus (1)

• Homotopy groups of complements to ample divisors

Cited by in corpus (6)

• On the irreducible components of characteristic varieties
• Hodge spectrum of hyperplane arrangements
• Katz's middle convolution algorithm
• Non vanishing loci of Hodge numbers of local systems
• Mixed multiplier ideals and the irregularity of abelian coverings of the projective plane
• The Chern character of a parabolic bundle, and a parabolic Reznikov theorem in the case of finite order at infinity