Locally conformal Kahler manifolds with potential

A locally conformally Kahler (LCK) manifold M is one which is covered by a Kahler manifold (M) over tilde with the deck transformation group acting conformally on (M) over tilde. If M admits a holomorphic flow, acting on (M) over tilde conformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vaisman manifolds is stable under small deformations. We define a new class of LCK-manifolds, called LCK manifolds with potential, which is closed under small deformations. All Vaisman manifolds are LCK with potential. We show that an LCK-manifold with potential admits a covering which can be compactified to a Stein variety by adding one point. This is used to show that any LCK manifold M with potential, dim M >= 3, can be embedded into a Hopf manifold, thus improving similar results for Vaisman manifolds Ornea and Verbitsky (Math Ann 332: 121-143, 2005).