Locally conformally Kaehler manifolds with potential

Details Locally conformally Kaehler manifolds with potential Locally conformally Kaehler manifolds with potential

2004-07-13

arxiv.org/abs/math/0407231v5

Mathematische Annalen, Vol. 248 (1), 2010, pp. 25-33

Mathematics

Algebraic Geometry

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

Scientific paper

A locally conformally K\"ahler (LCK) manifold $M$ is one which is covered by a K\"ahler 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.

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