In algebraic geometry, a complex manifold is called Fujiki class
[1] Let M be a compact manifold of Fujiki class
Moreover, the Douady space of X (that is, the moduli of deformations of a subvariety
[4] J.-P. Demailly and M. Pǎun have shown that a manifold is in Fujiki class
[5] They also conjectured that a manifold M is in Fujiki class
if it admits a nef current which is big, that is, satisfies For a cohomology class
which is rational, this statement is known: by Grauert-Riemenschneider conjecture, a holomorphic line bundle L with first Chern class nef and big has maximal Kodaira dimension, hence the corresponding rational map to is generically finite onto its image, which is algebraic, and therefore Kähler.
This conjecture was disproven in 1992 by Y.-S. Poon and Claude LeBrun [8]