Let
A
=
{
x
1
,
…
,
x
n
}
\mathcal {A} = \{x_1, \dotsc , x_n\}
be a subspace arrangement with a geometric lattice such that
codim
(
x
)
≥
2
\operatorname {codim}(x) \geq 2
for every
x
∈
A
x \in \mathcal {A}
. Using rational homotopy theory, we prove that the complement
M
(
A
)
M(\mathcal {A})
is rationally elliptic if and only if the sum
x
1
⊥
+
…
+
x
n
⊥
x_1^\perp + \dotso + x_n^\perp
is a direct sum. The homotopy type of
M
(
A
)
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
(
A
)
M(\mathcal {A})
is rationally elliptic, and show that most arrangements have a hyperbolic complement.