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

TL;DR

The paper addresses extending Hermite-Einstein and slope stability theory to a setting with an auxiliary form $Ω$ on a compact Kähler manifold, enabling $(ω,Ω)$-dependent notions. It develops generalized Hermite-Einstein metrics and a corresponding slope stability framework using the operator $P_{ω,Ω}$ and a modified curvature condition, establishing that $(ω,Ω)$-HE metrics imply $(ω,Ω)$-semi-stability and a holomorphic decomposition into $(ω,Ω)$-stable factors with the same Einstein constant. The main result extends Kobayashi–Lübke to this generalized context and underpins a coherent-sheaf theory with $(ω,Ω)$-degree and slope, including a generalized vanishing principle to deduce stability and splitting. Overall, the work broadens the scope of the Kobayashi-Hitchin correspondence to flexible geometric data, with potential implications for $m$-positivity and Monge-Ampère-type equations for forms of positive degree.

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.

Theorems & Definitions (15)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Corollary 2.4
• Proposition 2.5
• Proposition 2.6
• Lemma 2.7
• Theorem 2.8
• Lemma 2.9
• Lemma 2.10