Motivic zeta functions of hyperplane arrangements

TL;DR

This paper establishes a formula connecting motivic zeta functions of hyperplane arrangements to their Milnor fibers, revealing local constancy properties and providing combinatorial expressions related to characteristic polynomials.

Contribution

It introduces a new formula for motivic zeta functions of hyperplane arrangements in terms of Milnor fibers and characteristic polynomials, linking algebraic and combinatorial invariants.

Findings

The motivic zeta function can be expressed via Milnor fibers.

The Hodge-Deligne specialization is locally constant on realization spaces.

A combinatorial formula relates motivic Igusa zeta functions to characteristic polynomials.

Abstract

For each central essential hyperplane arrangement $\mathcal{A}$ over an algebraically closed field, let $Z_\mathcal{A}^{\hat\mu}(T)$ denote the Denef-Loeser motivic zeta function of $\mathcal{A}$. We prove a formula expressing $Z_\mathcal{A}^{\hat\mu}(T)$ in terms of the Milnor fibers of related hyperplane arrangements. We use this formula to show that the map taking each complex arrangement $\mathcal{A}$ to the Hodge-Deligne specialization of $Z_{\mathcal{A}}^{\hat\mu}(T)$ is locally constant on the realization space of any loop-free matroid. We also prove a combinatorial formula expressing the motivic Igusa zeta function of $\mathcal{A}$ in terms of the characteristic polynomials of related arrangements.