Solving the $\partial \overline{\partial}$ for extendable currents without vanishing the boundary cohomology group

TL;DR

This paper addresses solving the $ ext{ extbackslash dbar}$-equation with specified support in complex manifolds, providing solutions for extendable currents without requiring boundary cohomology vanishing.

Contribution

It introduces a method to solve the $ ext{ extbackslash dbar}$-equation for extendable currents without assuming boundary cohomology vanishing, expanding the applicability of such solutions.

Findings

Solved $ ext{ extbackslash dbar}$-equation with support in complex domains

Achieved solutions for extendable currents without boundary cohomology vanishing

Extended the understanding of cohomology in complex analysis

Abstract

In this paper, we consider the problem of solving the $\partial\overline{\partial}$ equation with discribed support for differential forms in a relatively compact domain $\Omega$ of a complex analytic manifold $X$. And as a consequence, we have the solution of the equation $\partial\overline{\partial}$ for extendable currents without the annulation assumption of the De Rham cohomology group of the boundary.\\ \textbf{Keywords:} operator $\partial\overline{\partial}$, De Rham cohomology group, Dolbeault cohomology group, Bott-Chern cohomology group,Applie cohomology group, discribed support, extendable currents.