[Paper Review] Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers

Nero Budur|Oct 11, 2006

Algebraic Geometry and Number Theory35 references14 citations

TL;DR

This paper establishes a structure theorem for unitary local systems on quasi-projective varieties using multiplier ideals and parabolic line bundles, proving polynomial periodicity of Hodge numbers $h^{q,0}$ for congruence covers and extending this to cohomology of unitary local systems. It generalizes results of Green-Lazarsfeld, Simpson, and Sarnak-Adams, and provides a geometric characterization of finite abelian covers, confirming a conjecture of Libgober on Hodge numbers of abelian covers.

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.

Motivation & Objective

• To develop a global framework for studying singularities of divisors in complex projective varieties using multiplier ideals and unitary local systems.
• To generalize the structure theorems of Green-Lazarsfeld and Simpson to the quasi-projective setting via complex tori and rational polytopes.
• To prove polynomial periodicity of Hodge numbers $h^{q,0}$ for congruence covers in any dimension, extending results of Hironaka and Sakuma.
• To extend polynomial periodicity to Betti numbers of unbranched congruence covers, generalizing Sarnak-Adams.
• To provide a geometric characterization of finite abelian covers, recovering and extending classical results of Pardini and others, and to confirm Libgober's conjecture on Hodge numbers of abelian covers.

Proposed method

• The paper uses parabolic line bundles to parametrize unitary local systems of rank one on the complement of a divisor in a complex projective variety.
• It introduces a stratification of the unitary local system space via multiplier ideals, showing these strata are finite unions of torsion translates of subtori in the Picard variety.
• The structure theorem is established by analyzing the cohomology of unitary local systems and using absolute functors and duality theorems from Saito and Simpson.
• The proof leverages the fact that the loci defined by cohomological dimension and Hodge filtration are absolute closed sets, hence their restrictions to unitary local systems are finite unions of torsion translates of subtori.
• The authors use the theory of logarithmic differentials and canonical extensions to compute the Hodge filtration on cohomology, enabling the analysis of $\text{Gr}_F^p H^m$.
• The periodicity is derived via a refinement of the decomposition of the boundary polytope $B(X,D)$, using the structure of $W^{p,q}_i(U,\mathcal{W})$ loci and their intersections.

Experimental results

Research questions

• RQ1 How can multiplier ideals be used to stratify the space of unitary local systems on the complement of a divisor in a complex projective variety?
• RQ2 What is the global structure of the moduli space of unitary local systems in the quasi-projective case, and how does it relate to complex tori and rational polytopes?
• RQ3 Does the Hodge number $h^{q,0}$ of congruence covers exhibit polynomial periodicity in any dimension, and if so, under what conditions?
• RQ4 Can the polynomial periodicity of Betti numbers of unbranched congruence covers be generalized beyond the original Sarnak-Adams result?
• RQ5 Is there a geometric characterization of finite abelian covers that recovers and extends the classical results of Pardini and others, and does it confirm Libgober's conjecture on Hodge numbers?

Key findings

• The space of unitary local systems on the complement of a divisor is stratified by multiplier ideals, and these strata are finite unions of torsion translates of subtori in the Picard variety.
• The structure theorem generalizes Green-Lazarsfeld and Simpson’s results to the quasi-projective case, showing that the cohomological loci are finite unions of torsion translates of subtori.
• Polynomial periodicity of Hodge numbers $h^{q,0}$ is established for congruence covers in any dimension, extending results of Hironaka and Sakuma.
• The polynomial periodicity extends to Betti numbers of unbranched congruence covers, generalizing the Sarnak-Adams theorem.
• A geometric characterization of finite abelian covers is derived, which recovers and extends the classical results of Pardini and confirms Libgober’s conjecture on Hodge numbers of abelian covers.
• The loci defined by the dimension of $\text{Gr}_F^p H^m$ for unitary local systems are shown to be finite unions of torsion translates of subtori in the unitary local system moduli space.