R\'esolution du $\partial\bar\partial$ pour les courants prolongeables d\'efinis dans un anneau

TL;DR

This paper addresses solving the $ar{ ext{d}}ar{ ext{d}}$ equation for extendable currents in complex domains, progressing from Euclidean space to complex manifolds and specific domains defined by plurisubharmonic functions.

Contribution

It extends the solvability of the $ar{ ext{d}}ar{ ext{d}}$ problem for currents from $ ext{C}^n$ to complex manifolds and particular domains, generalizing previous results.

Findings

Solved $ar{ ext{d}}ar{ ext{d}}$ for currents in $ ext{C}^n$ minus a ball

Extended the solution to contractible complex manifolds

Addressed the problem in domains defined by strictly plurisubharmonic functions

Abstract

Dans ce papier, on r\'esout d'abord le $\partial\bar\partial$ pour les courants prolongeables d\'efinis dans $\mathbb{C}^n$ priv\'e d'une boule $B$ de $\mathbb{C}^n$, ensuite dans une vari\'et\'e analytique complexe contractile $X$, enfin on le r\'esout pour un domaine $D=X\setminus\Omega$, o\`u $\Omega$ est un domaine born\'e de $X$ d\'efini par $\{z\in X\,\ /\,\ \varphi(z)<0\}$, (avec $\varphi$ une fonction d'exhaustion strictement plurisouharmonique). In this present paper, we first solve the $\partial\bar\partial$ for extendable currents definite in $\mathbb{C}^n\setminus B$, where $B$ is a ball of $\mathbb{C}^n$, then in a contractible analytic complex manifold $X$, and finally in a domain $D=X\setminus\Omega$ where $\Omega$ is a bounded domain of $X$ defined by $\{z\in X\,\ /\,\ \varphi(z)<0\}$, ($\varphi$ is an exhaution strictly plurisubharmonic function).