Let M be a compact manifold of Fujiki class C, and its complex subvariety. Then X is also in Fujiki class C (Lemma 4.6). Moreover, the Douady space of X (that is, the moduli of deformations of a subvariety , M fixed) is compact and in Fujiki class C.
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
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