  An $L^2$ version of the Serre duality on domains in complex manifolds
involving duality of Hilbert space realizations of the
$\bar{\partial}$-operator is established. This duality is used to study the
solution of the $\bar{\partial}$-equation with prescribed support. Applications
are given to $\bar{\partial}$-closed extension of forms, as well to
Bochner-Hartogs type extension of CR functions.
