Non-formality of Milnor fibers of line arrangements

TL;DR

This paper demonstrates that the Milnor fiber of a hyperplane arrangement can lack 1-formality, contrasting with the known formality of the arrangement's complement, through a specific example in dimension 2.

Contribution

It provides the first example showing the Milnor fiber of a hyperplane arrangement may not be 1-formal, challenging previous assumptions about their topological properties.

Findings

The complement of a hyperplane arrangement is formal.

The Milnor fiber associated to the arrangement may not be 1-formal.

An explicit example in dimension 2 illustrates this non-formality.

Abstract

The complement of a hyperplane arrangement in the complex projective space is known to be formal. We prove the global Milnor fiber associated to the homogeneous polynomial defining the arrangement may not even be 1-formal, by giving an example in dimension 2.