Existence of approximate Hermitian-Einstein structures on semistable principal bundles

TL;DR

This paper proves that a principal G-bundle over a compact Kähler manifold is semistable if and only if it admits approximate Hermitian-Einstein structures, establishing a key equivalence in complex differential geometry.

Contribution

It establishes the equivalence between semistability and the existence of approximate Hermitian-Einstein structures for principal G-bundles over Kähler manifolds.

Findings

Semistability of principal bundles is characterized by approximate Hermitian-Einstein structures.

The paper provides a necessary and sufficient condition linking geometric stability and differential geometric structures.

This result extends the understanding of Hermitian-Einstein metrics to principal bundles in complex geometry.

Abstract

Let E_G be a principal G-bundle over a compact connected K\"ahler manifold, where G is a connected reductive complex linear algebraic group. We show that E_G is semistable if and only if it admits approximate Hermitian-Einstein structures.