Koppelman formulas on flag manifolds

TL;DR

This paper develops explicit Koppelman formulas on flag manifolds and Grassmannians, enabling new proofs of vanishing theorems and constructing reproducing kernels for harmonic forms.

Contribution

It introduces Koppelman formulas for forms with values in various bundles on flag manifolds, providing novel tools for complex geometry and representation theory.

Findings

New explicit Koppelman formulas on flag manifolds

Simplified proofs of Bott-Borel-Weil vanishing theorems

Reproducing kernels for harmonic forms on Grassmannians

Abstract

We construct Koppelman formulas on manifolds of flags in $\C^N$ for forms with values in any holomorphic line bundle as well as in the tautological vector bundles and their duals. As an application we obtain new explicit proofs of some vanishing theorems of the Bott-Borel-Weil type by solving the corresponding $\debar$-equation. We also construct reproducing kernels for harmonic $(p,q)$-forms in the case of Grassmannians.