Balanced metrics and Gauduchon cone of locally conformally Kahler manifolds

A complex Hermitian n-manifold (M, I, ) is called locally conformally Kahler (LCK) if d=, where is a closed 1-form, balanced if ^n-1 is closed, and SKT if dId=0. We conjecture that any compact complex manifold admitting two of these three types of Hermitian forms (balanced, SKT, LCK) also admits a Kahler metric, and prove partial results towards this conjecture. We conjecture that the (1, 1) -form -d (I) is Bott--Chern homologous to a positive (1, 1) -current. This conjecture implies that (M, I) does not admit a balanced Hermitian metric. We verify this conjecture for all known classes of LCK manifolds.