The Bott Formula for Toric Varieties

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 ${\mathrm{\Omega }}_{\mathbb{P}}^p(D)={\mathrm{\Omega }}_{\mathbb{P}}^p\otimes {\mathcal{O}}_{\mathbb{P}}(D)$ of $p$-th differential forms 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 ${\mathcal{O}}_{\mathbb{P}}(D)$. Comparison of two versions of the Bott formula gives some elegant corollaries in the combinatorics of simple polytopes. Also, we obtain a generalization of the reciprocity law. Some applications of the Bott formula are discussed.