Degenerating Hermitian metrics and spectral geometry of the canonical bundle

TL;DR

This paper studies the spectral properties of the Dolbeault operator on the canonical bundle over singular Hermitian complex spaces, establishing key analytical features and applications to spectral geometry and heat operators.

Contribution

It introduces new analytical results on the Dolbeault operator's spectrum and properties on singular Hermitian spaces, with implications for spectral geometry and heat kernel analysis.

Findings

Closed range of the Dolbeault operator extension

Compactness of the domain inclusion

Discreteness of the spectrum of the Laplacian

Abstract

Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $m$. In this paper we are interested in the Dolbeault operator acting on the space of $L^2$ sections of the canonical bundle of $reg(X)$, the regular part of $X$. More precisely let $\overline{\mathfrak{d}}_{m,0}:L^2\Omega^{m,0}(reg(X),h)\rightarrow L^2\Omega^{m,1}(reg(X),h)$ be an arbitrarily fixed closed extension of $\overline{\partial}_{m,0}:L^2\Omega^{m,0}(reg(X),h)\rightarrow L^2\Omega^{m,1}(reg(X),h)$ where the domain of the latter operator is $\Omega_c^{m,0}(reg(X))$. We establish various properties such as closed range of $\overline{\mathfrak{d}}_{m,0}$, compactness of the inclusion $\mathcal{D}(\overline{\mathfrak{d}}_{m,0})\hookrightarrow L^2\Omega^{m,0}(reg(X),h)$ where $\mathcal{D}(\overline{\mathfrak{d}}_{m,0})$, the domain of $\overline{\mathfrak{d}}_{m,0}$, is endowed with the corresponding graph norm, and discreteness of the spectrum of the associated Hodge-Kodaira Laplacian $\overline{\mathfrak{d}}_{m,0}^*\circ \overline{\mathfrak{d}}_{m,0}$ with an estimate for the growth of its eigenvalues. Several corollaries such as trace class property for the heat operator associated to $\overline{\mathfrak{d}}_{m,0}^*\circ \overline{\mathfrak{d}}_{m,0}$, with an estimate for its trace, are derived. Finally in the last part we provide several applications to the Hodge-Kodaira Laplacian in the setting of both compact irreducible Hermitian complex spaces with isolated singularities and complex projective surfaces.