Zusammenfassung
A locally conformally Kahler (lcK) manifold is a complex manifold (M, J) together with a Hermitian metric g which is conformal to a Kahler metric in the neighbourhood of each point. In this paper we obtain three classification results in locally conformally Kahler geometry. The first one is the classification of conformal classes on compact manifolds containing two nonhomothetic Kahler metrics. ...
Zusammenfassung
A locally conformally Kahler (lcK) manifold is a complex manifold (M, J) together with a Hermitian metric g which is conformal to a Kahler metric in the neighbourhood of each point. In this paper we obtain three classification results in locally conformally Kahler geometry. The first one is the classification of conformal classes on compact manifolds containing two nonhomothetic Kahler metrics. The second one is the classification of compact Einstein locally conformally Kahler manifolds. The third result is the classification of the possible (restricted) Riemannian holonomy groups of compact locally conformally Kahler manifolds. We show that every locally (but not globally) conformally Kahler compact manifold of dimension 2n has holonomy SO(2n), unless it is Vaisman, in which case it has restricted holonomy SO(2n - 1). We also show that the restricted holonomy of a proper globally conformally Kahler compact manifold of dimension 2n is either SO(2n), or SO(2n - 1), or U(n), and we give the complete description of the possible solutions in the last two cases.
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