part 3 of the learning seminar

An intriguing open problem in complex geometry is to construct an example or prove nonexistence of a real analytic (or smooth), closed (i.e, compact without boundary), Levi-flat hypersurface in the complex projective plane $CP^2$. This question appeared in the context of foliation theory as a problem of the existence of nontrivial minimal sets. Nonexistence of Levi-flats in $CP^n$ for$ n > 2$ was proven by several authors, but the problem remains open for $n=2$. I will give the necessary background concerning Levi-flat hypersurfaces and outline three (mostly) self-contained proofs of the nonexistence results in $CP^n$ , $n > 2$. Two of the three proofs are due to Lins Neto, the third one is due to Siu. These proofs are genuinely different, and it is remarkable that all of them fail in dimension 2 for different reasons.