7 problems

Popovici's conjecture. Every compact ∂∂̄-manifold is balanced.

Let $(M^3,g)$ be a compact balanced Hermitian threefold, meaning that its Kähler form $\omega$ satisfies … Let $H$ denote the holomorphic sectional curvature of the Chern connectio…

A compact complex manifold satisfying the $\partial\bar{\partial}$-Lemma is called a $\partial\bar{\partial}$-manifold, and a compact complex manifold carrying a balanced metric is…

Conjecture on balanced metrics and the $\partial\overline{\partial}$-Lemma. The metric balanced condition and the cohomological $\partial\overline{\partial}$-Lemma property are str…

Fu–Xiao's conjecture.

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

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…