Hyperplane arrangements and mixed Hodge numbers of the Milnor fiber

TL;DR

This paper investigates the mixed Hodge structure of Milnor fibers of hyperplane arrangements using tropical geometry, providing invariance results and combinatorial descriptions of their topological and cohomological properties.

Contribution

It introduces a tropical geometric approach to study the Hodge-Deligne polynomial and boundary complex of Milnor fibers, revealing invariance and combinatorial structures.

Findings

Hodge-Deligne polynomial is locally constant on realization space

Provides a combinatorial description of the boundary complex homotopy type

Derives a formula for the top weight cohomology of Milnor fibers

Abstract

For each complex central essential hyperplane arrangement $\mathcal{A}$, let $F_{\mathcal{A}}$ denote its Milnor fiber. We use Tevelev's theory of tropical compactifications to study invariants related to the mixed Hodge structure on the cohomology of $F_{\mathcal{A}}$. We prove that the map taking each arrangement $\mathcal{A}$ to the Hodge-Deligne polynomial of $F_{\mathcal{A}}$ is locally constant on the realization space of any loop-free matroid. When $\mathcal{A}$ consists of distinct hyperplanes, we also give a combinatorial description for the homotopy type of the boundary complex of any simple normal crossing compactification of $F_{\mathcal{A}}$. As a direct consequence, we obtain a combinatorial formula for the top weight cohomology of $F_{\mathcal{A}}$, recovering a result of Dimca and Lehrer.