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

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

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