Topological computation of the first Milnor fiber cohomology of hyperplane arrangements

TL;DR

This paper introduces a topological method to compute the first Milnor fiber cohomology of hyperplane arrangements, establishing vanishing results for certain monodromy eigenspaces under connectivity conditions, and reduces the problem to line arrangements in projective planes.

Contribution

It provides a new topological approach to calculate Milnor fiber cohomology and proves vanishing of eigenspaces under specific geometric conditions, extending previous conjectures.

Findings

Vanishing of monodromy eigenspaces under connectivity assumptions

Reduction of the problem to line arrangements in P^2

Validation of conjectures for m ≥ 5

Abstract

We study a topological method to calculate the first Milnor fiber cohomology of a defining polynomial of a reduced projective hyperplane arrangement $X$ of degree $d$. We can show the vanishing of a monodromy eigenspace of the first Milnor fiber cohomology with eigenvalue of order $m\ge 2$ if $X\setminus(X^{[(m)]}\cup X^{\langle 3\rangle})$ or more generally $X\setminus(X^{[(m)]}\cup X^{\langle 3\rangle}\cup X_d)$ is connected. Here $X^{[(m)]}$ is the set of points of $X$ with multiplicity divisible by $m$, and $X^{\langle 3\rangle}:=\bigcup_{i,j,k}X_i\cap X_j\cap X_k$ with $X_i$ the irreducible components of $X$, where the union is taken over $i,j,k$ with ${\rm codim}\,X_i\cap X_j\cap X_k=3$. This hypothesis can be relaxed to some extent. The assertion is reduced to the case of a line arrangement in ${\bf P}^2$ by Artin's vanishing theorem (where $X^{\langle 3\rangle}=\emptyset$), and we use a projection from ${\bf P}^2$ to ${\bf P}^1$ with center a sufficiently general point of $X_d$. It may be expected that the assumption of an improved assertion is always satisfied for $m\ge 5$ (and also for $m=4$ except the Hessian arrangement). The resulting vanishing of eigenspaces has been conjectured for $m\ge 5$.