On toric locally conformally K\"ahler manifolds

TL;DR

This paper investigates the structure of compact toric locally conformally K"ahler manifolds, revealing their Kodaira dimension, classification of complex surfaces, and properties of associated Vaisman manifolds.

Contribution

It establishes the Kodaira dimension as -infinity for these manifolds and classifies the complex surfaces and Vaisman manifolds that admit such structures.

Findings

Kodaira dimension of underlying complex manifold is -infinity

Only diagonal Hopf surfaces admit toric strict locally conformally K"ahler metrics

Toric Vaisman manifolds have lcK rank 1 and are mapping tori of automorphisms

Abstract

We study compact toric strict locally conformally K\"ahler manifolds. We show that the Kodaira dimension of the underlying complex manifold is $-\infty$ and that the only compact complex surfaces admitting toric strict locally conformally K\"ahler metrics are the diagonal Hopf surfaces. We also show that every toric Vaisman manifold has lcK rank 1 and is isomorphic to the mapping torus of an automorphism of a toric compact Sasakian manifold.