A characterisation of toric LCK manifolds

Abstract

We prove that a compact toric locally conformally Kähler manifold which is not Kähler admits a toric Vaisman structure, a fact which was conjectured in mmp. This is the final step leading to the classification of compact toric locally conformally Kähler manifolds started in p and mmp. We also show, by constructing an example, that unlike in the symplectic case, toric locally conformally symplectic manifolds are not necessarily toric locally conformally Kähler.