Hodge-Deligne equivariant polynomials and monodromy of hyperplane arrangements

TL;DR

This paper explores the relationship between monodromy and mixed Hodge structures in hyperplane arrangement Milnor fibers, revealing new weight results and confirming spectrum determination by combinatorial data for line arrangements.

Contribution

It introduces new insights into the weights of hyperplane arrangement Milnor fibers and offers a novel proof that spectrum is determined by combinatorial data for line arrangements.

Findings

New weight results for hyperplane arrangement Milnor fibers

Spectrum determined by combinatorial data for line arrangements

Failure of Hodge-Deligne polynomial determination by combinatorial data

Abstract

We investigate the interplay between the monodromy and the Deligne mixed Hodge structure on the Milnor fiber of a homogeneous polynomial. In the case of hyperplane arrangement Milnor fibers, we obtain a new result on the possible weights. For line arrangements, we prove in a new way the fact due to Budur and Saito that the spectrum is determined by the weak combinatorial data, and show that such a result fails for the Hodge-Deligne polynomials.