Generalised Hermite-Einstein Fibre Metrics and Slope Stability for Holomorphic Vector Bundles

TL;DR

This paper generalizes the concepts of Hermite-Einstein metrics and slope stability for holomorphic vector bundles on compact complex manifolds, establishing a link between these notions in a broader geometric context.

Contribution

It introduces new definitions of Hermite-Einstein and stability conditions depending on additional forms, extending classical results to this more general setting.

Findings

Hermite-Einstein condition implies semi-stability.

Holomorphic bundles split into stable subbundles.

Extension of Kobayashi-Lübke theorem to new setting.

Abstract

Let $X$ be a compact complex manifold of dimension $n$ and let $m$ be a positive integer with $m\leq n$. Assume that $X$ admits a K\"ahler metric $\omega$ and a weakly positive, $\partial\bar\partial$-closed, smooth $(n-m,\,n-m)$-form $\Omega$. We introduce the notions of $(\omega,\,\Omega)$-Hermite-Einstein holomorphic vector bundles and $(\omega,\,\Omega)$(-semi)-stable coherent sheaves on $X$ by generalising the classical definitions depending only on $\omega$. We then prove that the $(\omega,\,\Omega)$-Hermite-Einstein condition implies the $(\omega,\,\Omega)$-semi-stability of a holomorphic vector bundle and its splitting into $(\omega,\,\Omega)$-stable subbundles. This extends a classical result by Kobayashi and L\"ubke to our generalised setting.