Balanced metrics and Gauduchon cone of locally conformally Kahler manifolds

A complex Hermitian $n$-manifold $(M,I,\omega )$ is called locally conformally Kahler (LCK) if $d\omega =\theta \wedge \omega$, where $\theta$ is a closed 1-form, balanced if ${\omega }^{n-1}$ is closed, and SKT if $dId\omega =0$. 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 $-d(I\theta )$ 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.