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.[1]

[edit] Properties

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

[edit] 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.[4] 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

c1(L) = [ω]

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[5] and Ueno[6] asked whether the property C is stable under deformations. This conjecture was disproven in 1992 by Y.-S. Poon and C. LeBrun[7]

[edit] References

  1. ^ A. Fujiki, "On Automorphism Groups of Compact Kähler Manifolds," Inv. Math. 44 (1978) 225-258. MR481142
  2. ^ A. Fujiki, Closedness of the Douady spaces of compact Kahler spaces, Publ. Res. Inst. Math. Sci. 14 (1978/79), no. 1, 1--52.MR486648
  3. ^ A. Fujiki, On the Douady space of a compact complex space in the category C. Nagoya Math. J. 85 (1982), 189--211.MR86j:32048
  4. ^ Demailly, Jean-Pierre; Paun, Mihai Numerical characterization of the Kahler cone of a compact Kahler manifold, Ann. of Math. (2) 159 (2004), no. 3, 1247--1274. MR2005i:32020
  5. ^ A. Fujiki, "On a Compact Complex Manifold in C without Holomorphic 2-Forms," Publ. RIMS 19 (1983). MR84m:32037
  6. ^ K. Ueno, ed., "Open Problems," Classification of Algebraic and Analytic Manifolds, Birkhaser, 1983.
  7. ^ Claude LeBrun, Yat-Sun Poon, "Twistors, Kahler manifolds, and bimeromorphic geometry II", J. Amer. Math. Soc. 5 (1992) MR92m:32053