Locally conformal Kähler manifolds with potential

Abstract

A locally conformally Kahler (LCK) manifold M is one which is covered by a Kahler manifold \({\widetilde M}\) with the deck transformation group acting conformally on \({\widetilde M}\). If M admits a holomorphic flow, acting on \({\widetilde M}\) 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, 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).