3 problems

Balanced metric conjecture. Each compact complex manifold $X$ satisfying the $\partial\overline{\partial}$-lemma admits a balanced metric.

Cone duality conjecture. The dual cone of $\mathcal E$ is the closure of the balanced cone:

Let $\pi:\mathcal{X}\to\Delta$ be a holomorphic family of compact complex manifolds, with fibres $X_t$. A compact complex manifold is balanced if it has the balanced property. Bala…