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

TL;DR

This paper establishes an $L^2$ Serre duality for domains in complex manifolds, enabling new insights into the $ar{ ext{-}} ext{d}$-equation solutions and extension problems for forms and CR functions.

Contribution

It introduces an $L^2$ Serre duality framework for complex domains, connecting Hilbert space realizations of the $ar{ ext{-}} ext{d}$-operator, and applies it to extension and solution problems.

Findings

Duality of Hilbert space realizations of $ar{ ext{-}} ext{d}$-operator established

Solutions to $ar{ ext{-}} ext{d}$-equation with prescribed support analyzed

Extensions of forms and CR functions demonstrated

Abstract

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.