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]
Properties
Let M be a compact manifold of Fujiki class
, and
its complex subvariety. Then Xis 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]
Conjectures
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
![{\displaystyle \int _{M}\omega ^{dim_{\mathbb {C} }M}>0.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
For a cohomology class
which is rational, this statement is known: by Grauert-Riemenschneider conjecture, a holomorphic line bundle L with first Chern class
![{\displaystyle c_{1}(L)=[\omega ]}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
nef and big has maximal Kodaira dimension, hence the corresponding rational map to
![{\displaystyle {\mathbb {P} }H^{0}(L^{N})}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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]
References
- ^ Fujiki, Akira (1978). "On Automorphism Groups of Compact Kähler Manifolds". Inventiones Mathematicae. 44 (3): 225–258. Bibcode:1978InMat..44..225F. doi:10.1007/BF01403162. MR 0481142.
- ^ 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. MR 0486648.
- ^ Fujiki, Akira (1982). "On the douady space of a compact complex space in the category C {\displaystyle {\mathcal {C}}}
". Nagoya Mathematical Journal. 85: 189–211. doi:10.1017/S002776300001970X. MR 0759679.
- ^ Angella, Daniele; Tomassini, Adriano (2013). "On the ∂ ∂ ¯ {\displaystyle \partial {\bar {\partial }}} -Lemma and Bott-Chern cohomology" (PDF). Inventiones Mathematicae. 192: 71–81. doi:10.1007/s00222-012-0406-3. S2CID 253747048.
- ^ 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 C {\displaystyle {\mathcal {C}}} without Holomorphic 2-Forms". Publications of the Research Institute for Mathematical Sciences. 19: 193–202. doi:10.2977/PRIMS/1195182983. MR 0700948.
- ^ 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