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×ℙm with values in various line bundles. Finally, we apply our method to developing weighted Koppelman formulas on Stein manifolds.
Citation
Elin Götmark. "Weighted integral formulas on manifolds." Ark. Mat. 46 (1) 43 - 68, April 2008. https://doi.org/10.1007/s11512-007-0056-7
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