Weighted integral formulas on manifolds

Abstract

We present a method of finding weighted Koppelman formulas for (p, q)-forms on n-dimensional complex manifolds X which admit a vector bundle of rank n over X×X, such that the diagonal of X×X has a defining section. We apply the method to ℙⁿ and find weighted Koppelman formulas for (p, q)-forms with values in a line bundle over ℙⁿ. As an application, we look at the cohomology groups of (p, q)-forms over ℙⁿ with values in various line bundles, and find explicit solutions to the $\overline{\partial}$-equation in some of the trivial groups. We also look at cohomology groups of (0, q)-forms over ℙⁿ×ℙ^m with values in various line bundles. Finally, we apply our method to developing weighted Koppelman formulas on Stein manifolds.