Conjectures
J.-P. Demailly and M. Paun have shown that a manifold is in Fujiki class C if and only if it supports a Kähler current. They also conjectured that a manifold M is in Fujiki class C 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.
Fujiki and Ueno asked whether the property C is stable under deformations. This conjecture was disproven in 1992 by Y.-S. Poon and C. LeBrun
Read more about this topic: Fujiki Class C
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