Maximal twisted Betti numbers of complex hyperplane arrangement complements

TL;DR

The paper proves that for an essential affine complex hyperplane arrangement $\mathcal{A}$ in $\mathbb{C}^n$ and a nontrivial rank $r$ local system $\mathcal{L}$ on the complement $U_\mathcal{A}$, the twisted Betti numbers satisfy $b_i(U_\mathcal{A};\mathcal{L}) < r\,b_i(U_\mathcal{A})$ for all $0\le i\le n$. Equality for some $i$ forces equality for all $i$ in this range and implies that $\mathcal{L}$ is the constant sheaf; the result is established by induction on the ambient dimension, with a reduction to central arrangements. The approach combines perverse-sheaf techniques, nearby cycles, and a Hopf fibration spectral sequence to analyze monodromy, together with Lefschetz hyperplane methods to handle noncentral cases. As a byproduct, the work extends the Brieskorn decomposition to arbitrary finite-rank local systems and provides a positive answer to a question of Yoshinaga and Liu for all complex arrangements.

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.

Theorems & Definitions (21)

• Theorem 1.1
• Corollary 1.2
• Proposition 1.3
• Remark 1.4
• Lemma 2.1
• proof
• proof : Proof of Proposition \ref{['prop surj']}
• Proposition 2.2
• proof
• Proposition 2.3