$L^{2}$ Serre duality on domains in complex manifolds and applications

Abstract

An L 2 version of the Serre duality on domains in complex manifolds involving duality of Hilbert space realizations of the @-operator is established. This duality is used to study the solution of the @-equation with prescribed support. Applications are given to @-closed extension of forms, as well to Bochner-Hartogs type extension of CR functions. comp (;E � ) with the quotient topology, where we endow spaces of compactly supported forms with the natural inductive limit topology. In fact, condition that the two maps in (1) have closed range is also necessary for the duality theorem to hold (see (9); also see (26, 27, 28) for further results of this type.) Serre's original proof (33) is based on sheaf theory and the theory of topological vector spaces. A different approach to this result, in the case when is a compact complex manifold, was given by Kodaira using Hodge theory (see (23) or (7).) In this note we extend Kodaira's method to non-compact Hermitian manifolds to obtain an L 2 analog of the Serre duality. Special cases of Serre-duality using L 2 methods have appeared before in many contexts (see (25), or (11, Theorem 5.1.7) and (19, 20), for example.) Our treatment aims to streamline and systematize these results, with emphasis on non-compact manifolds, and point out its close relation with the choice of L 2 -realizations of the Cauchy-Riemann operator @, or alternatively, choice of boundary conditions for the L 2 -realizations of the formal complex Laplacian @E#E + #E@E. The L 2 -duality can be interpreted in many ways. At one level, it is a duality between the standard � - Laplacian with @-Neumann boundary conditions, and thec-Laplacian with dual ( @-Dirichlet) boundary conditions. Using another approach, results regarding solution of the @-equation in L 2 can be converted to statements regarding the solution of the @c equation. This leads to a solution of the @-Cauchy problem, i.e., solution of the @-equation with prescribed support. At the heart of the matter lies the existence of