Fujiki class C

In algebraic geometry, a complex manifold is called Fujiki class $$\mathcal{C}$$ 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 $$\mathcal{C}$$, and $$X\subset M$$ its complex subvariety. Then X is also in Fujiki class $$\mathcal{C}$$ (, Lemma 4.6). Moreover, the Douady space of X (that is, the moduli of deformations of a subvariety $$X\subset M$$, M fixed) is compact and in Fujiki class $$\mathcal{C}$$.

Fujiki class $$\mathcal{C}$$ manifolds are examples of compact complex manifolds which are not necessarily Kähler, but for which the $\partial \bar \partial$-lemma holds.

Conjectures
J.-P. Demailly and M. Pǎun have shown that a manifold is in Fujiki class $$\mathcal{C}$$ if and only if it supports a Kähler current. They also conjectured that a manifold M is in Fujiki class $$\mathcal{C}$$ if it admits a nef current which is big, that is, satisfies


 * $$\int_M \omega^>0.$$

For a cohomology class $$[\omega]\in H^2(M)$$ which is rational, this statement is known: by Grauert-Riemenschneider conjecture, a holomorphic line bundle L with first Chern class


 * $$c_1(L)=[\omega]$$

nef and big has maximal Kodaira dimension, hence the corresponding rational map to


 * $${\mathbb P} H^0(L^N)$$

is generically finite onto its image, which is algebraic, and therefore Kähler.

Fujiki and Ueno asked whether the property $$\mathcal{C}$$ is stable under deformations. This conjecture was disproven in 1992 by Y.-S. Poon and Claude LeBrun