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August 26, 2025Open Access

Nonexistence of Homogeneous Levi-Flat Hypersurfaces in CP2

Key Points

No homogeneous levi-flat hypersurfaces exist in CP2, resolving part of an open question.
The study combines classification methods from homogeneous CR-manifolds and projective foliation geometry.
Techniques from complex geometry were employed to tackle the longstanding issue of their existence.
This finding indicates significant restrictions on the types of structures allowed in CP2's geometry.

Abstract

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.

Nonexistence of Homogeneous Levi-Flat Hypersurfaces in CP2

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