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

Let $X$ be a compact complex manifold of dimension $n$ and let $m$ be a positive integer with $m\le n$. Assume that $X$ admits a Kähler metric $\omega$ and a weakly positive, $\mathrm{\partial }\bar{\mathrm{\partial }}$-closed, smooth $(n-m,n-m)$-form $\mathrm{\Omega }$. We introduce the notions of $(\omega ,\mathrm{\Omega })$-Hermite-Einstein holomorphic vector bundles and $(\omega ,\mathrm{\Omega })$(-semi)-stable coherent sheaves on $X$ by generalising the classical definitions depending only on $\omega$. We then prove that the $(\omega ,\mathrm{\Omega })$-Hermite-Einstein condition implies the $(\omega ,\mathrm{\Omega })$-semi-stability of a holomorphic vector bundle and its splitting into $(\omega ,\mathrm{\Omega })$-stable subbundles. This extends a classical result by Kobayashi and Lübke to our generalised setting. In the appendix, we propose notions of both strongly and weakly (strictly) positive forms and currents and discuss their various properties.