Combinatorial Nonresonance Theorems for Hyperplane Arrangement Complements

TL;DR

This paper investigates nonresonance conditions for complex local systems on hyperplane arrangement complements, providing combinatorial criteria and extending existing theorems with new techniques.

Contribution

It refines existing methods to establish combinatorial sufficient conditions for nonresonance and strengthens related theorems by removing previous restrictions.

Findings

Established a combinatorial sufficient condition for nonresonance.

Strengthened a theorem of Bailet, Dimca, and Yoshinaga by removing a condition.

Developed restriction and lifting techniques for line arrangements.

Abstract

We study the nonresonance phenomenon for complex rank-one local systems on complements of hyperplane arrangements. We refine the method of Cohen, Dimca, and Orlik and obtain a combinatorial sufficient condition for nonresonance. As an application, we strengthen a theorem of Bailet, Dimca, and Yoshinaga by removing one of its conditions. We also develop restriction and lifting techniques to prove a nonresonance theorem for line arrangements.