$L^2$ Serre Duality on Domains in Complex Manifolds and Applications

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