Examples of non-trivial rank in locally conformal K\"ahler geometry

TL;DR

This paper characterizes the rank of locally conformal Kähler manifolds using equivariant geometry and algebraic number theory, providing the first explicit examples of non-trivial rank and analyzing specific Oeljeklaus-Toma manifolds.

Contribution

It introduces a new equivariant perspective on locally conformal Kähler geometry and constructs explicit examples with non-trivial rank using algebraic number theory.

Findings

The rank r of a locally conformal Kähler manifold can be non-trivial.

Explicit examples with 1<r<b_1 are constructed.

Oeljeklaus-Toma manifolds have rank either b_1 or b_1/2.

Abstract

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