Fujiki class C
In algebraic geometry, a complex manifold is called Fujiki class [math]\displaystyle{ \mathcal{C} }[/math] 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 [math]\displaystyle{ \mathcal{C} }[/math], and [math]\displaystyle{ X\subset M }[/math] its complex subvariety. Then X is also in Fujiki class [math]\displaystyle{ \mathcal{C} }[/math] (,[2] Lemma 4.6). Moreover, the Douady space of X (that is, the moduli of deformations of a subvariety [math]\displaystyle{ X\subset M }[/math], M fixed) is compact and in Fujiki class [math]\displaystyle{ \mathcal{C} }[/math].[3]
Fujiki class [math]\displaystyle{ \mathcal{C} }[/math] manifolds are examples of compact complex manifolds which are not necessarily Kähler, but for which the [math]\displaystyle{ \partial \bar \partial }[/math]-lemma holds.[4]
Conjectures
J.-P. Demailly and M. Pǎun have shown that a manifold is in Fujiki class [math]\displaystyle{ \mathcal{C} }[/math] if and only if it supports a Kähler current.[5] They also conjectured that a manifold M is in Fujiki class [math]\displaystyle{ \mathcal{C} }[/math] if it admits a nef current which is big, that is, satisfies
- [math]\displaystyle{ \int_M \omega^{{dim_{\mathbb C} M}}\gt 0. }[/math]
For a cohomology class [math]\displaystyle{ [\omega]\in H^2(M) }[/math] which is rational, this statement is known: by Grauert-Riemenschneider conjecture, a holomorphic line bundle L with first Chern class
- [math]\displaystyle{ c_1(L)=[\omega] }[/math]
nef and big has maximal Kodaira dimension, hence the corresponding rational map to
- [math]\displaystyle{ {\mathbb P} H^0(L^N) }[/math]
is generically finite onto its image, which is algebraic, and therefore Kähler.
Fujiki[6] and Ueno[7] asked whether the property [math]\displaystyle{ \mathcal{C} }[/math] is stable under deformations. This conjecture was disproven in 1992 by Y.-S. Poon and Claude LeBrun[8]
References
- ↑ Fujiki, Akira (1978). "On Automorphism Groups of Compact Kähler Manifolds". Inventiones Mathematicae 44 (3): 225–258. doi:10.1007/BF01403162. Bibcode: 1978InMat..44..225F. http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00209391X.
- ↑ Fujiki, Akira (1978). "Closedness of the Douady spaces of compact Kähler spaces". Publications of the Research Institute for Mathematical Sciences 14: 1–52. doi:10.2977/PRIMS/1195189279.
- ↑ Fujiki, Akira (1982). "On the douady space of a compact complex space in the category [math]\displaystyle{ \mathcal{C} }[/math].". Nagoya Mathematical Journal 85: 189–211. doi:10.1017/S002776300001970X.
- ↑ Angella, Daniele; Tomassini, Adriano (2013). "On the [math]\displaystyle{ \partial \bar \partial }[/math] -Lemma and Bott-Chern cohomology". Inventiones Mathematicae 192: 71–81. doi:10.1007/s00222-012-0406-3. http://eprints.adm.unipi.it/1612/1/angella%2Dtomassini%2Drevised.pdf.
- ↑ Demailly, Jean-Pierre; Pǎun, Mihai Numerical characterization of the Kahler cone of a compact Kahler manifold, Ann. of Math. (2) 159 (2004), no. 3, 1247--1274. MR2113021
- ↑ Fujiki, Akira (1983). "On a Compact Complex Manifold in [math]\displaystyle{ \mathcal{C} }[/math] without Holomorphic 2-Forms". Publications of the Research Institute for Mathematical Sciences 19: 193–202. doi:10.2977/PRIMS/1195182983.
- ↑ K. Ueno, ed., "Open Problems," Classification of Algebraic and Analytic Manifolds, Birkhaser, 1983.
- ↑ Claude LeBrun, Yat-Sun Poon, "Twistors, Kahler manifolds, and bimeromorphic geometry II", J. Amer. Math. Soc. 5 (1992) MR1137099
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