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

$\int _{M}\omega ^{dim_{\mathbb {C} }M}>0.$

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

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

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