Locally conformally Kaehler manifolds with potential

Abstract

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

arXiv comment14 pages, v. 5: section about the embedding of Sasakian manifolds eliminated due to an error