In this article, we study the range of the Cauchy-Riemann operator \({\overline{\partial }}\) on domains in the complex projective space \({\mathbb{C}\mathbb{P}}^n\) . In particular, we show that \({\overline{\partial }}\) does not have closed range in \(L^2\) for (2,1)-forms on the Hartogs triangle in \({\mathbb{C}\mathbb{P}}^2\) . We also study the \({\overline{\partial }}\) -Cauchy problem on pseudoconvex domains and use it to prove the Sobolev estimates for \({\overline{\partial }}\) on pseudoconcave domains in \({\mathbb{C}\mathbb{P}}^n\) .
