Maximal twisted Betti numbers of complex hyperplane arrangement complements

TL;DR

This paper proves that the Betti numbers of local systems on complex hyperplane arrangement complements are maximized only when the local system is constant, resolving a recent open question.

Contribution

It establishes a precise condition under which Betti numbers are maximized, answering a question posed by Yoshinaga and the first author.

Findings

Betti numbers are maximized only for constant local systems.

The result applies to essential complex hyperplane arrangements.

It provides a definitive criterion for maximal Betti numbers.

Abstract

We show that the Betti numbers of a local system on the complement of an essential complex hyperplane arrangement are maximized precisely when the local system is constant. This result answers positively a recent question of Yoshinaga and the first author.