A characterisation of toric locally conformally Kähler manifolds

We prove that a compact toric locally conformally K¨ahler manifold which is not K¨ahler admits a toric Vaisman structure. This is the final step leading to the classification of compact toric locally conformally K¨ahler manifolds. We also show, by constructing an example, that unlike in the symplectic case, toric locally conformally symplectic manifolds are not necessarily toric locally conformally K¨ahler.