Solving the $\partial \overline{\partial}$ for extendable currents without vanishing the boundary cohomology group
Mamadou Eramane Bodian, Souhaibou Sambou, Sény Diatta, Salomon Sambou
TL;DR
The paper tackles the $\partial\overline{\partial}$-problem with described support on relatively compact domains in complex manifolds, aiming to avoid vanishing boundary cohomology assumptions. It develops a direct Bott-Chern cohomology method on Stein and analytic manifolds, showing that closed $(1,1)$-forms with compact support are $\partial\overline{\partial}$-exact under certain compactly supported cohomology vanishing conditions, via decomposing $d=g_1+g_2$ and employing pluriharmonic extensions. The authors extend the approach to general $(p,q)$-forms, establishing vanishing of $H^{p,q}_{BC,\overline{\Omega}}(X)$ under suitable hypotheses and, by duality, obtain extendable-current solutions for $\partial\overline{\partial}$-equations with prescribed support. This work removes the previous necessity of boundary cohomology vanishing, and yields corollaries for completely pseudoconvex domains, broadening the solvability of the $\partial\overline{\partial}$ problem in the setting of extendable currents.
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 $Ω$ 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.