Solving the $\partial \bar{\partial}$ with prescribed support

TL;DR

This paper addresses the problem of solving the $ar{ ext{partial}}$-$ar{ ext{partial}}$ equation with specific support constraints within complex manifolds, advancing methods for complex analysis and differential forms.

Contribution

It introduces new techniques for solving the $ar{ ext{partial}}$-$ar{ ext{partial}}$ equation with prescribed support in complex manifolds, filling a gap in existing mathematical theory.

Findings

Developed a method for solving $ar{ ext{partial}}$-$ar{ ext{partial}}$ with support constraints

Extended existing theory to broader classes of forms and currents

Provided conditions for solvability in complex domains

Abstract

In this paper, we consider the problem of solving the $\partial \bar{\partial}$ with prescribed support for forms or currents in a domain $\Omega$ of an complex manifold $X$.