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

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