Fujiki class C

In algebraic geometry, a complex manifold is called Fujiki class ${\displaystyle {\mathcal {C}}}$ if it is bimeromorphic to a compact Kähler manifold. This notion was defined by Akira Fujiki.^[1]

Properties

Let M be a compact manifold of Fujiki class ${\displaystyle {\mathcal {C}}}$, and ${\displaystyle X\subset M}$ its complex subvariety. Then X is also in Fujiki class ${\displaystyle {\mathcal {C}}}$ (,^[2] Lemma 4.6). Moreover, the Douady space of X (that is, the moduli of deformations of a subvariety ${\displaystyle X\subset M}$, M fixed) is compact and in Fujiki class ${\displaystyle {\mathcal {C}}}$.^[3]

Fujiki class ${\displaystyle {\mathcal {C}}}$ manifolds are examples of compact complex manifolds which are not necessarily Kähler, but for which the ${\displaystyle \partial {\bar {\partial }}}$-lemma holds.^[4]

Conjectures

J.-P. Demailly and M. Pǎun have shown that a manifold is in Fujiki class ${\displaystyle {\mathcal {C}}}$ if and only if it supports a Kähler current.^[5] They also conjectured that a manifold M is in Fujiki class ${\displaystyle {\mathcal {C}}}$ if it admits a nef current which is big, that is, satisfies

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

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

${\displaystyle c_{1}(L)=[\omega ]}$

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

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

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

Fujiki^[6] and Ueno^[7] asked whether the property ${\displaystyle {\mathcal {C}}}$ is stable under deformations. This conjecture was disproven in 1992 by Y.-S. Poon and Claude LeBrun ^[8]