Balanced metrics and Gauduchon cone of locally conformally Kahler manifolds

A complex Hermitian n-manifold (M,I, ω) is called locally conformally Kahler (LCK) if dω=θω, where θ is a closed 1-form, balanced if ωⁿ⁻¹ is closed, and SKT if dIdω=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(Iθ) 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.