  Let A be a subspace arrangement with a geometric lattice such that codim(x) >
1 for every x in A. Using rational homotopy theory, we prove that the
complement M(A) is rationally elliptic if and only if the sum of the orthogonal
subspaces is a direct sum. The homotopy type of M(A) is also given: it is a
product of odd dimensional spheres. Finally, some other equivalent conditions
are given, such as Poincare duality. Those results give a complete description
of arrangements (with geometric lattice and with the codimension condition on
the subspaces) such that M(A) is rationally elliptic, and show that most
arrangements have an hyperbolic complement.
