Milnor fibers of real line arrangements

TL;DR

This paper introduces a new algorithm for computing monodromy eigenspaces of Milnor fibers associated with complexified real line arrangements, leveraging chamber adjacency to analyze topological properties and generalize existing results.

Contribution

The authors present a novel algorithm based on chamber adjacency for computing monodromy eigenspaces, extending previous vanishing results and characterizing specific arrangements.

Findings

Generalized vanishing results for Milnor monodromy

Provided new upper bounds for eigenspaces

Characterized the $A_3$-arrangement via monodromy

Abstract

We study Milnor fibers of complexified real line arrangements. We give a new algorithm computing monodromy eigenspaces of the first cohomology. The algorithm is based on the description of minimal CW-complexes homotopic to the complements, and uses the real figure, that is, the adjacency relations of chambers. It enables us to generalize a vanishing result of Libgober, give new upper-bounds and characterize the $A_3$-arrangement in terms of non-triviality of Milnor monodromy.