Campana–Demailly–Verbitsky conjecture on simple compact Kähler manifolds

Let $X$ be a simple compact Kähler manifold, meaning that a very general point of $X$ is not contained in any proper positive-dimensional analytic subvariety. Equivalently, $X$ admits no covering family of positive-dimensional analytic subvarieties.

Campana–Demailly–Verbitsky conjecture. Either $X$ has a finite étale cover which is bimeromorphic to a complex torus, or $H^0(X,\Omega_X^2)$ is generated by a holomorphic $2$-form $\sigma$ which is generically symplectic; that is, $\dim X=2m$ is even and

$$\sigma^{\wedge m}\neq 0$$

generically. In particular, one should have $\kappa(X)=0$. If $\dim X$ is odd, then $X$ should be bimeromorphic to a complex torus, possibly after passing to a finite étale cover.

This is a conjectural classification of simple compact Kähler manifolds by the standard irreducible pieces of Kähler geometry. Its three-dimensional case was proved, while the general statement remains open.