Koppelman Formulas on Smooth Compact Toric Varieties

In this paper we derive an explicit Koppelman integral representation formula in terms of the combinatorial data on smooth compact toric varieties for (0, q) smooth forms taking values in specific line bundles. The n-dimensional toric varieties are such that their Newton polyhedron contains the origin and the standard base {e(1), ..., e(n)} of R-n. Applying the above formula one obtains an alternative proof about vanishing of the Dolbeault cohomology groups of (0, q) forms over such smooth compact toric varieties with values in various lines bundles.