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

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 Kähler metric $ω$ and a weakly positive, $\partial\bar\partial$-closed, smooth $(n-m,\,n-m)$-form $Ω$. We introduce the notions of $(ω,\,Ω)$-Hermite-Einstein holomorphic vector bundles and $(ω,\,Ω)$(-semi)-stable coherent sheaves on $X$ by generalising the classical definitions depending only on $ω$. We then prove that the $(ω,\,Ω)$-Hermite-Einstein condition implies the $(ω,\,Ω)$-semi-stability of a holomorphic vector bundle and its splitting into $(ω,\,Ω)$-stable subbundles. This extends a classical result by Kobayashi and Lübke to our generalised setting.