Motivic zeta functions of hyperplane arrangements

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.

25 pages