Properties

Let M be a compact manifold of Fujiki class C, and $Xsubset\; M$ 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 $Xsubset\; M$, M fixed) is compact and in Fujiki class C.

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

$int\_M\; omega^\{\{dim\_\{Bbb\; C\}\; M\}\}>0.$

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

$\{Bbb\; P\}\; H^0(L^N)$

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

References