Girish Mahajan (Editor)

Fujiki class C

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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.

Contents

Properties

Let M be a compact manifold of Fujiki class C, and X ⊂ 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 X ⊂ M , M fixed) is compact and in Fujiki class C.

Conjectures

J.-P. Demailly and M. Pǎun 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

∫ M ω d i m C M > 0.

For a cohomology class [ ω ] ∈ 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 ) = [ ω ]

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

P H 0 ( L N )

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 Claude LeBrun

References

Fujiki class C Wikipedia
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