Solving $\overline\partial$ with prescribed support on Hartogs triangles in $\mathbb C^2$ and $\mathbb C\mathbb P^2$

TL;DR

This paper investigates the solvability of the $ar{ ext{d}}$ equation with prescribed support on Hartogs triangles and domains, revealing unexpected phenomena and the failure of Dolbeault isomorphism in these settings.

Contribution

It demonstrates the failure of Dolbeault isomorphism for cohomology with prescribed support on Hartogs domains in complex spaces, a novel insight in several complex variables.

Findings

Failure of Dolbeault isomorphism with prescribed support

Unexpected phenomena in solvability of $ar{ ext{d}}$ on Hartogs domains

New insights into cohomology in complex analysis

Abstract

In this paper we study the solvability of the Cauchy-Riemann equation with prescibed support in different spaces of forms. The unbounded Hartogs triangle in $\mathbb C^2$ and the Hartogs domains in $\mathbb C\mathbb P^2$ provide us new unexpected phenomena. In particular we prove that the Dolbeault isomorphism fails to hold for the Dolbeault cohomology with prescribed support