## Asian Journal of Mathematics

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

#### Abstract

The purpose of this paper is to investigate canonical metrics on a semi-stable vector bundle $E$ over a compact Kähler 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 Kähler case. As an application some basic properties of semi-stable vector bundles over compact Kähler manifolds are established, such as the fact that semi-stability is preserved under certain exterior and symmetric products.

#### Article information

Source
Asian J. Math., Volume 18, Number 5 (2014), 859-884.

Dates
First available in Project Euclid: 2 December 2014

https://projecteuclid.org/euclid.ajm/1417489245

Mathematical Reviews number (MathSciNet)
MR3287006

Zentralblatt MATH identifier
1315.53079

Subjects
Primary: 53XX 35XX

#### Citation

Jacob, Adam. Existence of approximate Hermitian-Einstein structures on semi-stable bundles. Asian J. Math. 18 (2014), no. 5, 859--884. https://projecteuclid.org/euclid.ajm/1417489245