Degenerating Hermitian metrics, canonical bundle and spectral convergence

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 $(A,h|_A)$.