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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]].<ref>A. Fujiki, ''"On Automorphism Groups of Compact Kähler Manifolds,"'' Inv. Math. 44 (1978) 225-258. {{MathSciNet | id = 481142}}</ref>

== Properties ==

Let ''M'' be a compact manifold of Fujiki class C, and
<math>X\subset M</math> its complex subvariety. Then ''X''
is also in Fujiki class C (,<ref>A. Fujiki, ''Closedness of the Douady spaces of compact Kahler spaces'', Publ. Res. Inst. Math. Sci. 14 (1978/79), no. 1, 1--52.{{MathSciNet | id = 486648}}</ref> Lemma 4.6). Moreover, the [[Douady space]] of ''X'' (that is, the [[moduli space|moduli of deformations]] of a subvariety <math>X\subset M</math>, ''M'' fixed) is compact and in Fujiki class C.<ref>A. Fujiki, ''On the Douady space of a compact complex space in the category C.'' Nagoya Math. J. 85 (1982), 189--211.{{MathSciNet | id = 86j:32048}}</ref>

== 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]].<ref>Demailly, Jean-Pierre; Paun, Mihai [http://arxiv.org/abs/math.AG/0105176 ''Numerical characterization of the Kahler cone of a compact Kahler manifold''], Ann. of Math. (2) 159 (2004), no. 3, 1247--1274. {{MathSciNet | id = 2005i:32020}}</ref>
They also conjectured that a manifold ''M'' is in Fujiki class C if it admits a [[nef current]] which is ''big'', that is, satisfies

:<math>\int_M \omega^{{dim_{\Bbb C} M}}>0.</math>

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

:<math>c_1(L)=[\omega]</math>

[[Numerically effective|nef]] and big has maximal [[Kodaira dimension]], hence the corresponding rational map to

:<math>{\Bbb P} H^0(L^N)</math>

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

Fujiki<ref>A. Fujiki, ''"On a Compact Complex Manifold in C without Holomorphic 2-Forms,"'' Publ. RIMS 19 (1983). {{MathSciNet | id = 84m:32037}}</ref> and Ueno<ref>K. Ueno, ed., ''"Open Problems,"'' Classification of Algebraic and Analytic Manifolds, Birkhaser, 1983.</ref> asked whether the property C is stable under deformations. This conjecture was disproven in 1992 by Y.-S. Poon and [[Claude LeBrun ]]<ref>[[Claude LeBrun]], Yat-Sun Poon, [http://arxiv.org/abs/alg-geom/9202006 ''"Twistors, Kahler manifolds, and bimeromorphic geometry II"''], J. Amer. Math. Soc. 5 (1992) {{MathSciNet | id = 92m:32053}}</ref>

==References==

<references/>

[[Category:Algebraic geometry]]
[[Category:Complex manifolds]]

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