On the non-existence of smooth Levi-flat hypersurfaces in $\mathbb{CP}_n$

Abstract

We prove that there exists no $C^m$ Levi-flat real hypersurface in $\mathbb{CP}_n$ for $n \geqslant 2$ and $m \geqslant 4$. This is an improvement of the regularity in a theorem of Y.-T. Siu who proved this result for $m \geqslant 8$.