Degenerating Hermitian metrics and spectral geometry of the canonical bundle

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 <(partial derivative)overbar>(m,0) : L-2 Omega(m,0)(reg(X), h) -> L-2 Omega(m,1)(reg(X), h) be an arbitrarily fixed closed extension of (partial derivative) over bar (m,0) : L-2 Omega(m,0)(reg(X), h) -> L-2 Omega(m,1)(reg(X), h) where the domain of the latter operator is Omega(m,0)(c)(reg(X)). We establish various properties such as closed range of (partial derivative) over bar (m,0) compactness of the inclusion D(<(partial derivative)overbar>(m,0) ->) L-2 Omega(m,0)(reg(X), h) where D(<(partial derivative)overbar>(m,0)), the domain of <(partial derivative)overbar>(m,0) is endowed with the corresponding graph norm, and discreteness of the spectrum of the associated Hodge-Kodaira Laplacian (partial derivative) over bar (m,0)* circle <(partial derivative)overbar>(m,0), with an estimate for the growth of its eigenvalues. Several corollaries such as trace class property for the heat operator associated to <(partial derivative)overbar>(m,0)* circle <(partial derivative)overbar>(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. (C) 2018 Elsevier Inc. All rights reserved.