On Formality and Combinatorial Formality for hyperplane arrangements

TL;DR

This paper explores the concept of formality in hyperplane arrangements, establishing that niceness implies formality and that formality is hereditary, contrasting with known non-hereditary conditions.

Contribution

It proves that niceness guarantees formality and that formality is hereditary, providing new insights into the properties of hyperplane arrangements.

Findings

Niceness of arrangements implies formality.

Formality is hereditary under restrictions.

$k$-formality is not hereditary.

Abstract

A hyperplane arrangement is called formal provided all linear dependencies among the defining forms of the hyperplanes are generated by ones corresponding to intersections of codimension two. The significance of this notion stems from the fact that complex arrangements with aspherical complements are formal. The aim of this note is twofold. While work of Yuzvinsky shows that formality is not combinatorial, in our first main theorem we prove that the combinatorial property of niceness of arrangements does entail formality. Our second main theorem shows that formality is hereditary, i.e. is passed to restrictions. This is rather counter-intuitive, as in contrast the known sufficient conditions for formality, i.e. asphericity, freeness and niceness (owed to our first theorem), are not hereditary themselves. We also demonstrate that the stronger property of $k$-formality, due to Brandt and Terao, is not hereditary.