Gauduchon metrics and Hermite-Einstein metrics on non-Kähler varieties

TL;DR

This work extends the theory of canonical metrics to non-Kähler and singular settings by constructing a bounded Gauduchon metric on any compact Hermitian variety via resolution methods and a desingularization path. It develops uniform Sobolev and Poincaré inequalities on K-domains to handle complex Monge–Ampère equations in singular contexts, enabling the definition of slope stability using Bott–Cchern/Aeppli data and the subsequent existence and uniqueness of singular Hermite–Einstein metrics for slope-stable reflexive sheaves on non-Kähler normal varieties. The paper also proves an openness property for stability under perturbations of Gauduchon metrics and establishes uniform a priori bounds (Harnack and $L^{ty}$-estimates) for approximate HE metrics, allowing a robust limiting process to obtain singular HE metrics on the regular locus. Collectively, these results generalize the Donaldson–Uhlenbeck–Yau correspondence to singular, non-Kähler settings, with potential implications for moduli spaces and non-Kähler geometry. The techniques combine resolution of singularities, pluripotential theory on singular spaces, and elliptic PDE methods tailored to Hermitian (non-Kähler) geometry.

Abstract

We show the existence of Gauduchon metrics on arbitrary compact hermitian varieties, generalizing our previous work on smoothable singularities. These metrics allow us to define the notion of slope stability for torsion-free coherent sheaves on compact normal varieties that are not necessarily Kähler. Then we prove the existence and uniqueness of singular Hermite-Einstein metrics for slope-stable reflexive sheaves on non-Kähler normal varieties.

Theorems & Definitions (58)

• Theorem A
• Theorem A': cf. Theorem \ref{['thm:existence_bdd_Gauduchon']}
• Theorem B: cf. Theorem \ref{['thm:unif_Sobolev_Poincare']}
• Theorem C
• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Definition 2.1
• Definition 2.2