Examples of non-trivial rank in locally conformal Kähler geometry

Abstract

We consider locally conformal Kähler geometry as an equivariant, homothetic Kähler geometry (K, Γ). We show that the de Rham class of the Lee form can be naturally identified with the homomorphism projecting Γ to its dilation factors, thus completing the description of locally conformal Kähler geometry in this equivariant setting. The rank r _M of a locally conformal Kähler manifold is the rank of the image of this homomorphism. Using algebraic number theory, we show that r _M is non-trivial, providing explicit examples of locally conformal Kähler manifolds with ${1\nless{\text{\upshape \rmfamily r}_{M}}\nless b_1}$. As far as we know, these are the first examples of this kind. Moreover, we prove that locally conformal Kähler Oeljeklaus-Toma manifolds have either r _M = b ₁ or r _M = b ₁/2.