The Monodromy Conjecture for hyperplane arrangements

TL;DR

This paper proves the weak Monodromy Conjecture for hyperplane arrangements, linking poles of zeta functions to monodromy eigenvalues, and reduces the strong version to a specific conjecture about roots of Bernstein-Sato polynomials.

Contribution

It establishes the weak Monodromy Conjecture for hyperplane arrangements and reduces the strong version to a new conjecture involving Bernstein-Sato polynomial roots.

Findings

Proves the weak Monodromy Conjecture for hyperplane arrangements.

Reduces the strong conjecture to a specific root conjecture for Bernstein-Sato polynomials.

Provides a new perspective on the relationship between poles and monodromy eigenvalues.

Abstract

The Monodromy Conjecture asserts that if c is a pole of the local topological zeta function of a hypersurface, then exp(2\pi i c) is an eigenvalue of the monodromy on the cohomology of the Milnor fiber. A stronger version of the conjecture asserts that every pole is a root of the Bernstein-Sato polynomial of the hypersurface. In this note we prove the weak version of the conjecture for hyperplane arrangements. Furthermore, we reduce the strong version to the following conjecture: -n/d is always a root of the Bernstein-Sato polynomial of an indecomposable essential central hyperplane arrangement of d hyperplanes in the affine n-space.