11 problems

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

Let $M$ be a compact complex manifold of real dimension $2n$ with vanishing first Chern class. A hermitian structure on $M$ is a Riemannian metric compatible with its complex struc…

Let $M$ be a minimal surface of Kodaira dimension $-\infty$ with $b_2>0$ and $b_1=1$. A real 3-sphere $S\subset M$ is a smoothly embedded 3-sphere, and $M\backslash S$ denotes its…

$p$-SKT/$p$-Kähler conjecture.

Gauduchon-to-balanced hyperbolicity conjecture. If $X$ admits a balanced metric $omega$ which is Gauduchon hyperbolic, then $X$ is balanced hyperbolic.

Let $M$ be a compact complex manifold with a complex structure $J$. An LCK metric and a pluriclosed metric on $M$ are said to be compatible with the same complex structure when bot…

Let $X$ be a compact Calabi–Yau threefold with holomorphic volume form $\Omega$, and let $\mathfrak{b}_0$ be a balanced class on $X$. Let $V$ be a holomorphic vector bundle over…

Let $X$ be a compact complex surface. A non-Kähler complex surface is one that admits no Kähler metric, and a birational structure is a complex atlas with transition maps in the Cr…

A compact complex surface is called non-Kähler if it does not admit a Kähler metric, and a birational structure is an atlas whose transition maps lie in the Cremona group…

A Hermitian metric is balanced when its fundamental form is balanced. An astheno-Kähler metric is a Hermitian metric $[?]$ satisfying, in complex dimension $n$, $[?]$; in complex d…

A balanced manifold is a complex manifold admitting a Hermitian metric whose fundamental form is balanced; a Calabi–Yau manifold has holomorphically trivial canonical bundle. A Bot…