$L^2$-Sobolev Theory for $\bar{\mathrm{\partial }}$ on Domains in ${\mathbb{C}\mathbb{P}}^n$

In this article, we study the range of the Cauchy-Riemann operator $\bar{\mathrm{\partial }}$ on domains in the complex projective space ${\mathbb{C}\mathbb{P}}^n$. In particular, we show that $\bar{\mathrm{\partial }}$ 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 $\bar{\mathrm{\partial }}$-Cauchy problem on pseudoconvex domains and use it to prove the Sobolev estimates for $\bar{\mathrm{\partial }}$ on pseudoconcave domains in ${\mathbb{C}\mathbb{P}}^n$.