  An n-dimensional polytope P^n is called simple if exactly n codimension-one
faces meet at each vertex. The lattice of faces of a simple polytope P^n with m
codimension-one faces defines an arrangement of even-dimensional planes in
R^{2m}. We construct a free action of the group R^{m-n} on the complement of
this arrangement. The corresponding quotient is a smooth manifold Z_P invested
with a canonical action of the compact torus T^m with the orbit space P^n. For
each smooth projective toric variety M^{2n} defined by a simple polytope P^n
with the given lattice of faces there exists a subgroup T^{m-n}\subset T^m
acting freely on Z_P such that Z_P/T^{m-n}=M^{2n}. We calculate the cohomology
ring of Z_P and show that it is isomorphic to the cohomology ring of the face
ring of P^n regarded as a module over the polynomial ring. In this way the
cohomology of Z_P acquires a bigraded algebra structure, and the additional
grading allows to catch the combinatorial invariants of the polytope. At the
same time this gives an example of explicit calculation of the cohomology of
the complement of an arrangement of planes, which is of independent interest.
