In algebraic geometry, a complex manifold is called Fujiki class if it is bimeromorphic to a compact Kähler manifold. This notion was defined by Akira Fujiki. [1]
Let M be a compact manifold of Fujiki class , and its complex subvariety. Then X is also in Fujiki class (, [2] Lemma 4.6). Moreover, the Douady space of X (that is, the moduli of deformations of a subvariety , M fixed) is compact and in Fujiki class . [3]
Fujiki class manifolds are examples of compact complex manifolds which are not necessarily Kähler, but for which the -lemma holds. [4]
J.-P. Demailly and M. Pǎun have shown that a manifold is in Fujiki class if and only if it supports a Kähler current. [5] They also conjectured that a manifold M is in Fujiki class if it admits a nef current which is big, that is, satisfies
For a cohomology class which is rational, this statement is known: by Grauert-Riemenschneider conjecture, a holomorphic line bundle L with first Chern class
nef and big has maximal Kodaira dimension, hence the corresponding rational map to
is generically finite onto its image, which is algebraic, and therefore Kähler.
Fujiki [6] and Ueno [7] asked whether the property is stable under deformations. This conjecture was disproven in 1992 by Y.-S. Poon and Claude LeBrun [8]
In algebraic geometry, a complex manifold is called Fujiki class if it is bimeromorphic to a compact Kähler manifold. This notion was defined by Akira Fujiki. [1]
Let M be a compact manifold of Fujiki class , and its complex subvariety. Then X is also in Fujiki class (, [2] Lemma 4.6). Moreover, the Douady space of X (that is, the moduli of deformations of a subvariety , M fixed) is compact and in Fujiki class . [3]
Fujiki class manifolds are examples of compact complex manifolds which are not necessarily Kähler, but for which the -lemma holds. [4]
J.-P. Demailly and M. Pǎun have shown that a manifold is in Fujiki class if and only if it supports a Kähler current. [5] They also conjectured that a manifold M is in Fujiki class if it admits a nef current which is big, that is, satisfies
For a cohomology class which is rational, this statement is known: by Grauert-Riemenschneider conjecture, a holomorphic line bundle L with first Chern class
nef and big has maximal Kodaira dimension, hence the corresponding rational map to
is generically finite onto its image, which is algebraic, and therefore Kähler.
Fujiki [6] and Ueno [7] asked whether the property is stable under deformations. This conjecture was disproven in 1992 by Y.-S. Poon and Claude LeBrun [8]