An immersion theorem for Vaisman manifolds

Abstract

A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering Open image in new window, with monodromy acting on Open image in new window by Kahler homotheties. A compact LCK manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on Open image in new window. We prove that any compact Vaisman manifold admits a natural holomorphic immersion to a Hopf manifold (ℂn∖0)ℤ. As an application, we obtain that any Sasakian manifold has a contact immersion to an odd-dimensional sphere.