On the global structure of Hopf hypersurfaces in a complex space form

It is known that a tube over a Kahler submanifold in a complex space form i s a Hopf hypersurface. In some sense the reverse statement is true: a conne cted compact generic immersed C2n-1 regular Hopf hypersurface in the comple x projective space is a tube over an irreducible algebraic variety. In the complex hyperbolic space a connected compact generic immersed C2n-1 regular Hopf hypersurface is a geodesic hypersphere.