Immersion theorem for Vaisman manifolds

TL;DR

This paper proves that compact Vaisman manifolds can be holomorphically immersed into Hopf manifolds, extending classical embedding theorems, and shows Sasakian manifolds can be contact immersed into spheres, revealing new geometric relationships.

Contribution

It establishes a non-Kaehler analogue of Kodaira's embedding theorem for Vaisman manifolds and demonstrates contact immersions of Sasakian manifolds into spheres.

Findings

Compact Vaisman manifolds admit holomorphic immersions into Hopf manifolds.

Any Sasakian manifold can be contact immersed into an odd-dimensional sphere.

Provides new links between complex, contact, and Sasakian geometries.

Abstract

A locally conformally Kaehler (LCK) manifold is a complex manifold admitting a Kaehler covering M, with monodromy acting on M by Kaehler homotheties. A compact LCK manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on M. We prove a non-Kaehler analogue of Kodaira embedding theorem: any compact Vaisman manifold admits a natural holomorphic immersion to a Hopf manifold. As an application, we obtain that any Sasakian manifold has a contact immersion to an odd-dimensional sphere.