# Journal of the European Mathematical Society

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**Volume 22, Issue 1, 2020, pp. 119–149**

**DOI: 10.4171/JEMS/918**

Published online: 2019-09-16

Conformally related Kähler metrics and the holonomy of lcK manifolds

Farid Madani, Andrei Moroianu^{[1]}and Mihaela Pilca

^{[2]}(1) Université Paris-Sud, Orsay, France

(2) Universität Regensburg, Germany

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

*Keywords: *lcK structure, conformally Kähler structure, holonomy, Calabi Ansatz, Kähler structure, ambikähler structure

Madani Farid, Moroianu Andrei, Pilca Mihaela: Conformally related Kähler metrics and the holonomy of lcK manifolds. *J. Eur. Math. Soc.* 22 (2020), 119-149. doi: 10.4171/JEMS/918