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. [A. Fujiki, "On Automorphism Groups of Compact Kähler Manifolds," Inv. Math. 44 (1978) 225-258. MathSciNet | id = 481142]

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 ( [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] , 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. [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 ]

Conjectures

J.-P. Demailly and M. Paun haveshown that a manifold is in Fujiki class C if and onlyif it supports a Kähler current. [ 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] They also conjectured that a manifold "M" is in Fujiki class C if it admits a nef current which is "big", that is, satisfies

:$int\_M\; omega^$dim_{Bbb C} M>0.

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

:$c\_1(L)=\; [omega]$

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

:$\{Bbb\; P\}\; H^0(L^N)$

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

Fujiki [A. Fujiki, "On a Compact Complex Manifold in C without Holomorphic 2-Forms," Publ. RIMS 19 (1983). MathSciNet | id = 84m:32037] and Ueno [K. Ueno, ed., "Open Problems," Classification of Algebraic and Analytic Manifolds, Birkhaser, 1983.] asked whether the property C is stable under deformations. This conjecture was disproven in 1992 by Y.-S. Poon and C. LeBrun [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]

References