Torus actions and combinatorics of polytopes

Abstract

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.