Degenerating Hermitian metrics, canonical bundle and spectral convergence

TL;DR

This paper studies how spectral properties of certain Laplacians on a complex manifold behave as the underlying Hermitian metrics degenerate, establishing convergence results to a limit operator on a possibly singular space.

Contribution

It proves spectral convergence theorems for Hodge-Kodaira Laplacians under metric degeneration, extending understanding of spectral stability in complex geometry.

Findings

Eigenvalues converge to those of the limit operator

Heat kernels and heat operators exhibit convergence

Results apply to degenerations of Hermitian metrics on complex manifolds

Abstract

Let $(M,J)$ be a compact complex manifold of complex dimension $m$ and let $g_s$ be a one-parameter family of Hermitian forms on $M$ that are smooth and positive definite for each fixed $s\in (0,1]$ and that somehow degenerates to a Hermitian pseudometric $h$ for $s$ tending to $0$. In this paper under rather general assumptions on $g_s$ we prove various spectral convergence type theorems for the family of Hodge-Kodaira Laplacians $\Delta_{\overline{\partial},m,0,s}$ associated to $g_s$ and acting on the canonical bundle of $M$. In particular we show that, as $s$ tends to zero, the eigenvalues, the heat operators and the heat kernels corresponding to the family $\Delta_{\overline{\partial},m,0,s}$ converge to the eigenvalues, the heat operator and the heat kernel of $\Delta_{\overline{\partial},m,0,\mathrm{abs}}$, a suitable self-adjoint operator with entirely discrete spectrum defined on the limit space $(A,h|_A)$.