Cohen–Dimca–Orlik type conjecture for integer local systems on complex arrangements

Let $d4d0$ be a complex hyperplane arrangement in $d539^{\ell}$, not necessarily a complexified real arrangement, and let $M(\mathcal{A})$ denote its complement. Let $\mathcal{L}$ be a rank one $\mathbb{Z}$-local system on $M(\mathcal{A})$ satisfying the CDO-condition. Define

$$\beta_k=\left|\sum_{i=0}^{k}(-1)^i b_i(M(\mathcal{A}))\right|.$$

Cohen–Dimca–Orlik type conjecture. The cohomology groups satisfy

$$H^{k}(M(\mathcal{A}),\mathcal{L})\cong \begin{cases} \mathbb{Z}^{\beta_{\ell}}\oplus\mathbb{Z}_2^{\beta_{\ell-1}} & k=\ell,\\ \mathbb{Z}_2^{\beta_{k-1}} & 1\leq k\leq\ell-1,\\ 0 & \text{otherwise}, \end{cases}$$

and hence $H^{*}(M(\mathcal{A}),\mathcal{L})$ is combinatorially determined. For complexified real arrangements, the analogous assertion is proved in the paper; the conjecture extends it to arbitrary complex arrangements.