7 problems

Vanishing-property conjecture. Every big class satisfies the vanishing property.

Bott–Chern positivity conjecture. The $(1,1)$-form $-d(I\theta)$ is Bott–Chern homologous to a positive $(1,1)$-current.

Guedj–Lu's volume criterion. The class $\alpha$ contains a Kähler current if and only if

Let $X$ be a compact complex manifold of dimension $n$, and let $H^{1,1}_{BC}(X,\mathbb C)$ denote its complex Bott–Chern $(1,1)$-cohomology. A class is nef if it lies in the nef c…

Let $X$ be a compact complex manifold of dimension $n$, and let $H^{1,1}_{BC}(X,\mathbb R)$ denote its real Bott–Chern $(1,1)$-cohomology. A class $\alpha$ is nef if it lies in the…

The Hodge conjecture in Bott-Chern cohomology. Every element of $H^{p,p}_{BC}(Y;\Q)$ is representable by holomorphic $p$-chains with $\Q$-coefficients.

Vanishing Bott–Chern class conjecture. The Bott–Chern class of $M$ always vanishes.