First Milnor cohomology of hyperplane arrangements

TL;DR

This paper provides combinatorial and non-combinatorial formulas to determine the dimension of the non-unipotent part of the first Milnor cohomology in hyperplane arrangements, advancing understanding of their topological properties.

Contribution

It introduces new formulas for calculating the Milnor cohomology dimension, extending previous results and incorporating arrangement singularity positions.

Findings

A combinatorial lower bound formula for Milnor cohomology dimension.

Exact dimension formula under stronger combinatorial conditions.

A generalized non-combinatorial formula depending on singular point positions.

Abstract

We show a combinatorial formula for a lower bound of the dimension of the non-unipotent monodromy part of the first Milnor cohomology of a hyperplane arrangement satisfying some combinatorial conditions. This gives exactly its dimension if a stronger combinatorial condition is satisfied. We also prove a non-combinatorial formula for the dimension of the non-unipotent part of the first Milnor cohomology, which apparently depends on the position of the singular points. The latter generalizes a formula previously obtained by the second named author.