Existence of approximate Hermitian-Einstein structures on semi-stable bundles

TL;DR

This paper proves that semi-stable vector bundles over compact Kahler manifolds admit approximate Hermitian-Einstein structures, extending classical results from projective to Kahler geometry, and explores their stability properties under various bundle operations.

Contribution

It generalizes Kobayashi's classic result by establishing the existence of approximate Hermitian-Einstein structures on semi-stable bundles over Kahler manifolds.

Findings

Donaldson's functional is bounded from below for semi-stable bundles

Semi-stability is preserved under tensor, exterior, and symmetric products

Semi-stable bundles admit approximate Hermitian-Einstein structures

Abstract

The purpose of this paper is to investigate canonical metrics on a semi-stable vector bundle E over a compact Kahler manifold X. It is shown that, if E is semi-stable, then Donaldson's functional is bounded from below. This implies that E admits an approximate Hermitian-Einstein structure, generalizing a classic result of Kobayashi for projective manifolds to the Kahler case. As an application some basic properties of semi-stable vector bundles over compact Kahler manifolds are established, such as the fact that semi-stability is preserved under tensor product and certain exterior and symmetric products.