  For a Hermitian holomorphic vector bundle over a Hermitian manifold, we
consider the Dolbeault Laplacian with $\overline\partial$-Neumann boundary
conditions, which is a self-adjoint operator on the space of square-integrable
differential forms with values in the given holomorphic bundle. We argue that
some known results on the spectral properties of this operator on pseudoconvex
domains in $\mathbb C^n$ continue to hold on K\"ahler manifolds satisfying
certain bounded geometry assumptions. In particular, we will consider the
Dolbeault complex for forms with values in a line bundle, where known results
from magnetic Schr\"odinger operator theory can be applied.
