Conformally related K\"ahler metrics and the holonomy of lcK manifolds

Abstract

A locally conformally K\"ahler (lcK) manifold is a complex manifold $(M,J)$ together with a Hermitian metric $g$ which is conformal to a K\"ahler metric in the neighbourhood of each point. In this paper we obtain three classification results in locally conformally K\"ahler geometry. The first one is the classification of conformal classes on compact manifolds containing two non-homothetic K\"ahler metrics. The second one is the classification of compact Einstein locally conformally K\"ahler manifolds. The third result is the classification of the possible (restricted) Riemannian holonomy groups of compact locally conformally K\"ahler manifolds. We show that every locally (but not globally) conformally K\"ahler 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 K\"ahler 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.