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

Abstract

It is known that a tube over a Kähler submanifold in a complex space form is a Hopf hypersurface. In some sense the reverse statement is true: a connected compact generic immersed $C^{2n-1}$ regular Hopf hypersurface in the complex projective space is a tube over an irreducible algebraic variety. In the complex hyperbolic space a connected compact generic immersed $C^{2n-1}$ regular Hopf hypersurface is a geodesic hypersphere.