Solving the $\partial \bar{\partial}$ with prescribed support
Winnie Ossete Ingoba, Souhaibou Sambou, Salomon Sambou
TL;DR
The paper develops a framework to solve the $\partial\bar{\partial}$ equation with prescribed spatial support on domains inside complex manifolds. It leverages cohomological vanishing and pluriharmonic extension principles to establish both uniqueness properties for solutions and constructive existence results across multiple regularity classes (smooth, $L^p_{loc}$, $C^k$, and distribution). Central to the approach is the analysis of cohomology groups with support (and compact support) and the extension of pluriharmonic functions under cohomological hypotheses, yielding prescribed-support potentials for $(1,1)$-forms and currents. These results bridge de Rham and Dolbeault cohomology vanishing with Bott-Chern-type conclusions, enabling explicit solutions with compact support in a variety of settings, including currents in non-compact manifolds with suitable vanishing. The methods provide a unified path from abstract cohomological conditions to concrete solvability of $\partial\bar{\partial}$ with prescribed support, with implications for cohomology theories and pluriharmonic extension.
Abstract
In this paper, we consider the problem of solving the $\partial \bar{\partial}$ with prescribed support for forms or currents in a domain $Ω$ of an complex manifold $X$.