The Bott Formula for Toric Varieties

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The purpose of this paper is to give an explicit formula which allows one to compute the dimension of the cohomology groups of the sheaf Ω¶ᵖ(D) of p-th differential forms of Zariski twisted by an ample invertible sheaf on a complete simplicial toric variety. The formula involves some combinatorial sums of integer points over all faces of the support polytope for Øₓ(D). We also introduce a new combinatorial object, the so-called p-th Hilbert-Erhart polynomial, which generalizes the usual notion and behaves similar. Namely, there exists a generalization of the inversion law for a usual Hilbert-Erhart polynomial. Some applications of the Bott formula are discussed.