An immersion theorem for Vaisman manifolds

A locally conformally Kähler (LCK) manifold is a complex manifold admitting a Kähler covering , with monodromy acting on by Kähler homotheties. A compact LCK manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on . We prove that any compact Vaisman manifold admits a natural holomorphic immersion to a Hopf manifold (ⁿ0). As an application, we obtain that any Sasakian manifold has a contact immersion to an odd-dimensional sphere.Mathematics Subject Classification (2000): 53C55, 14E25, 53C25Liviu Ornea is member of EDGE, Research Training Network HRPN-CT-2000-00101, supported by the European Human Potential Programme.Misha Verbitsky is an EPSRC advanced fellow supported by CRDF grant RM1-2354-MO02 and EPSRC grant GR/R77773/01.Both authors acknowledge financial support from Ecole Polytechnique (Palaiseau).