Nonexistence of Homogeneous Levi-Flat Hypersurfaces in CP2

We investigate the longstanding question of whether compact Levi-flat hypersurfaces exist in the complex projective plane CP2. While the nonexistence of closed real-analytic Levi-flat hypersurfaces in CPn for n>2 is well known, the case n=2 remains open. By combining techniques from the classification of homogeneous CR-manifolds with projective foliation geometry, we prove that no homogeneous Levi-flat hypersurfaces exist in CP2, thus partially resolving the problem under natural symmetry assumptions.