# LCK metrics on toric LCS manifolds

## TL;DR

This paper establishes a classification of compact toric locally conformally symplectic manifolds with compatible complex structures via cone and scalar data, and shows these metrics admit positive potentials.

## Contribution

It provides a bijective correspondence between such manifolds and specific geometric data, and proves the existence of positive potentials for these metrics.

## Key findings

- Classification of manifolds via cones and scalar parameters
- Existence of positive potentials for toric locally conformally Kähler metrics
- Bijective correspondence between geometric data and manifolds

## Abstract

We show a bijective correspondence between compact toric locally conformally symplectic manifolds which admit a compatible complex structure and pairs $(C,a)$, where $C$ is a good cone in the dual Lie algebra of the torus and $a$ is a positive real number. Moreover, we prove that any toric locally conformally K\"ahler metric on a compact manifold admits a positive potential.