Definitions

# Fujiki class C

In algebraic geometry, a complex manifold is called Fujiki class 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 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

$int_M omega^\left\{\left\{dim_\left\{Bbb C\right\} M\right\}\right\}>0.$

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

$c_1\left(L\right)=\left[omega\right]$

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

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

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

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