  Let $\Omega$ be a pseudoconvex domain with $C^2$-smooth boundary in $\mathbb
CP^n$. We prove that the $\bar\partial-Neumann operator $N$ exists for
$(p,q)$-forms on $\Omega$. Furthermore, there exists a $t_0>0$ such that the
operators $N$, $\bar\partial^*N$, $\bar\partial N$ and the Bergman projection
are regular in the Sobolev space $W^t (\bar{\Omega}) $ for $t<t_0$.
  The boundary estimates above have applications in complex geometry. We use
the estimates to prove the nonexistence of $C^{2, \alpha}$ real Levi-flat
hypersurfaces in $\mathbb CP^n$. We also show that there exist no non-zero
$L^2$-holomorphic $(p, 0)$-forms on any pseudoconcave domain in $\mathbb CP^n$
with $p > 0$.
