Rational homotopy type of subspace arrangements with a geometric lattice

Let $\mathcal{A} = \{x_1, \dotsc, x_n\}$ be a subspace arrangement with a geometric lattice such that $\codim(x) \geq 2$ for every $x \in\mathcal{A}$. Using rational homotopy theory, we prove that the complement $M(\mathcal{A})$ is rationally elliptic if and only if the sum $x_1^\perp + \dotso + x_n^\perp$ is a direct sum. The homotopy type of $M(\mathcal{A})$ is also given: it is a product of odd-dimensional spheres. Finally, some other equivalent conditions are given, such as Poincaré duality. Those results give a complete description of arrangements (with a geometric lattice and with the codimension condition on the subspaces) such that $M(\mathcal{A})$ is rationally elliptic, and show that most arrangements have a hyperbolic complement.