Fujiki class C

In algebraic geometry, a complex manifold is called Fujiki class if it is bimeromorphic to a compact Kähler manifold. This notion was defined by Akira Fujiki.

Properties

Let M be a compact manifold of Fujiki class , and
its complex subvariety. Then X
is also in Fujiki class . Moreover, the Douady space of X is compact and in Fujiki class.
Fujiki class manifolds are examples of compact complex manifolds which are not necessarily Kähler, but for which the -lemma holds.

and M. Pǎun have
shown that a manifold is in Fujiki class if and only
if it supports a Kähler current.
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.
Fujiki and Ueno asked whether the property is stable under deformations. This conjecture was disproven in 1992 by Y.-S. Poon and Claude LeBrun