1999/09/28 by Victor M. Buchstaber, Taras E. Panov
Mathematics · #math.AT #math.AG #math.CO #msc:57R19 #msc:57S25 #msc:14M25 #msc:52B05
published as Proceedings of the Steklov Institute of Mathematics 225 (1999), 87-120 · 36 pages, LaTeX 2e, to appear in Proceedings of the Steklov Institute of Mathematics vol. 225
arxiv created 1999/09/28 · arxiv updated 2009/11/30
An n-dimensional polytope Pn is called simple if exactly n codimension-one faces meet at each vertex. The lattice of faces of a simple polytope Pn with m codimension-one faces defines an arrangement of even-dimensional planes in R2m. We construct a free action of the group Rm-n on the complement of this arrangement. The corresponding quotient is a smooth manifold ZP invested with a canonical action of the compact torus Tm with the orbit space Pn. For each smooth projective toric variety M2n defined by a simple polytope Pn with the given lattice of faces there exists a subgroup Tm-n⊂ Tm acting freely on ZP such that ZP/Tm-n=M2n. We calculate the cohomology ring of ZP and show that it is isomorphic to the cohomology ring of the face ring of Pn regarded as a module over the polynomial ring. In this way the cohomology of ZP 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.