Maximal Betti number for local system cohomology of hyperplane arrangement complements

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 ${\mathbb{C}}^{\mathrm{\ell }}$. Dimca-Papadima and Randell independently showed that $M(\mathcal{A})$ is homotopy equivalent to a minimal CW-complex. It implies that $\dim H^k(M(\mathcal{A}),\mathcal{L})\le b_k(M(\mathcal{A}))$. In this paper, we show that if $\mathcal{A}$ is real, then the inequality holds as equality for some $0\le k\le \mathrm{\ell }$ 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.