Maximal Betti number for local system cohomology of hyperplane arrangement complements

Abstract

Let $\mathcal{L}$ be a rank one local system with field coefficient on the complement $M(\mathcal{A})$ of an essential complex hyperplane arrangement $\mathcal{A}$ in $\mathbb{C}^\ell$. Dimca-Papadima and Randell independently showed that $M(\mathcal{A})$ is homotopy equivalent to a minimal CW-complex. It implies that $\dim H^k(M(\mathcal{A}),\mathcal{L}) \leq b_k(M(\mathcal{A}))$. In this paper, we show that if $\mathcal{A}$ is real, then the inequality holds as equality for some $0\leq k\leq \ell$ if and only if $\mathcal{L}$ is the constant sheaf. The proof is using the descriptions of local system cohomology of $M(\mathcal{A})$ in terms of chambers.

Figures (1)

• Figure 1: Opposite chambers

Theorems & Definitions (13)

• Theorem 1.1
• Proposition 1.4
• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Proposition 2.4
• Definition 2.5
• Proposition 2.6
• Theorem 2.7
• Lemma 3.1