Hyperplane arrangements with non-formal Milnor fibers

TL;DR

This paper establishes a combinatorial criterion based on multinet structures to identify hyperplane arrangements with non-1-formal Milnor fibers, providing new examples and insights into their topology.

Contribution

It introduces a sufficient combinatorial condition for non-1-formality of Milnor fibers and constructs an infinite family of arrangements exhibiting this property.

Findings

Identifies a multinet-based criterion for non-1-formality.

Constructs an infinite family of arrangements with non-formal Milnor fibers.

Provides background on cohomology jump loci and Milnor fiber topology.

Abstract

Each complex hyperplane arrangement $\mathcal{A}$ gives rise to a Milnor fibration of its complement. Building on work of Zuber, we give a combinatorial sufficient condition for the Milnor fiber $F(\mathcal{A})$ to be non-$1$-formal, expressed in terms of the multinet structure on $\mathcal{A}$, and use it to produce an infinite family of monomial arrangements $\mathcal{A}(3k,3k,3)$ with non-formal Milnor fibers. We also review the relevant background on cohomology jump loci, formality, and the topology of Milnor fibers of arrangements.