An analog of the Hard Lefschetz theorem for convex polytopes simple in edges

Proofs of these relations may be found in [1, 2, 3]. If the polytope ∆ is not simple, then the relations above are not true. A polytope ∆ is said to be integral provided all its vertices belong to the integer lattice. With each integral convex polytope ∆ one associates the toric variety X = X(∆) (see [4, 5, 6, 7]). This is a projective complex algebraic variety, singular in general. It turns out that the intersection cohomology Betti numbers (see [8, 9, 10]) of the toric vairety X are combinatorial invariants of the polytope ∆ (they are calculated in [11, 12]). For example, for simple polytope ∆ we have dim IH(X,C) = hk. The relations above on the h-vector of a simple convex polytope can be deduced from the Poincaré duality and the Hard Lefschetz theorem in the cohomology of X . Each simple polytope is combinatorially equivalent to an integral one. Therefore the study of the h-vector of a simple polytope reduces formally to the study of the intersection cohomology of toric varieties.