  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
