Abstract:
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 $\Omega_{\mathbb P}^p(D)= \Omega_{\mathbb P}^p\otimes {\mathcal O_\mathbb P}(D)$ 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.
Key words and phrases:$p$-th Hilbert–Ehrhart polynomial, Zariski forms.