Locally conformally Kaehler manifolds with potential

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A locally conformally Kähler (LCK) manifold M is one which is covered by a Kähler manifold M with the deck transform group acting conformally on M. If M admits a holomorphic flow, acting on M 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, M > 2, can be embedded to a Hopf manifold, thus improving on similar results for Vaisman. manifolds.