Singular Gauduchon metrics

TL;DR

The paper extends Gauduchon metric theory to irreducible compact singular hermitian varieties that admit a smoothing. It proves uniform bounds for Gauduchon factors on all smooth fibers in a smoothing family and constructs a bounded Gauduchon metric on the central fiber's regular part as a limit of nearby fibers. It further shows that the extended $(n-1,n-1)$-form is pluriclosed and establishes a uniqueness property for Gauduchon metrics in the singular setting. The results provide canonical, conformally stable Gauduchon structures in the singular/non-Kähler regime and supply a framework for extending Gauduchon-type geometry across deformations of singular spaces.

Abstract

In 1977, Gauduchon proved that on every compact hermitian manifold $(X, ω)$ there exists a conformally equivalent hermitian metric $ω_{\mathrm{G}}$ which satisfies $\mathrm{dd}^c ω_{\mathrm{G}}^{n-1} = 0$. In this note, we extend this result to irreducible compact singular hermitian varieties which admit a smoothing.

Theorems & Definitions (15)

• Theorem A: cf. Corollary \ref{['cor:G bdd family']} and Theorem \ref{['thm:Gau current']}
• Theorem A
• Definition 1.1
• Remark 1.1
• Definition 1.2
• Theorem 2.1
• Proposition 2.2
• Remark 2.1
• Corollary 2.3
• Lemma 2.4