Campana's classification conjecture for simple compact Kähler manifolds

Let $X$ be a simple compact Kähler manifold of dimension $n$, and write $q=h^1(X,\mathcal O_X)$ and $\kappa(X)$ for its irregularity and Kodaira dimension. A holomorphic $2$-form $\sigma$ on $X$ is generically symplectic when $n=2m$ is even and

$$\bigwedge^m\sigma\neq 0.$$

An irreducible hyperkähler manifold is a hyperkähler manifold with finite fundamental group and $h^{2,0}(X)=1$. Campana's classification conjecture. The following should hold: (1) either $X$ has a finite étale cover bimeromorphic to a complex torus, or $H^0(X,\Omega_X^2)$ is generated by a generically symplectic form $\sigma$; consequently $\kappa(X)=0$, and if $\dim X$ is odd then $X$ should be bimeromorphic to a complex torus, possibly after a finite étale cover; (2) if $X$ contains no nontrivial subvarieties, then $X$ should be either a complex torus or an irreducible hyperkähler manifold, with $K_X$ trivial and the two cases distinguished by $q>0$ and $q=0$; (3) if $X$ is generically symplectic, then $\pi_1(X)$ should be finite of cardinality at most $2^m$, where $2m=n$. These assertions form a proposed higher-dimensional analogue of the classification of simple surfaces: the known examples are general complex tori, $K3$ surfaces, and general members of hyperkähler deformation families. The classification is open in dimensions at least three; the cited examples show that general deformations of Hilbert schemes of $K3$ surfaces can have no subvarieties, while generalized Kummer varieties exhibit additional subvarieties.