  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 $(A,h|_A)$.
