Fujiki class C

In algebraic geometry, a complex manifold is called Fujiki class ${\displaystyle {\mathcal {C}}}$ 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 ${\displaystyle {\mathcal {C}}}$ , and ${\displaystyle X\subset M}$  its complex subvariety. Then X is also in Fujiki class ${\displaystyle {\mathcal {C}}}$  (,^[2] Lemma 4.6). Moreover, the Douady space of X (that is, the moduli of deformations of a subvariety ${\displaystyle X\subset M}$ , M fixed) is compact and in Fujiki class ${\displaystyle {\mathcal {C}}}$ .^[3]

Fujiki class ${\displaystyle {\mathcal {C}}}$  manifolds are examples of compact complex manifolds which are not necessarily Kähler, but for which the ${\displaystyle \partial {\bar {\partial }}}$ -lemma holds.^[4]

Conjectures

J.-P. Demailly and M. Pǎun have shown that a manifold is in Fujiki class ${\displaystyle {\mathcal {C}}}$  if and only if it supports a Kähler current.^[5] They also conjectured that a manifold M is in Fujiki class ${\displaystyle {\mathcal {C}}}$  if it admits a nef current which is big, that is, satisfies

${\displaystyle \int _{M}\omega ^{dim_{\mathbb {C} }M}>0.}$ 

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

${\displaystyle c_{1}(L)=[\omega ]}$ 

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

${\displaystyle {\mathbb {P} }H^{0}(L^{N})}$ 

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

Fujiki^[6] and Ueno^[7] asked whether the property ${\displaystyle {\mathcal {C}}}$  is stable under deformations. This conjecture was disproven in 1992 by Y.-S. Poon and Claude LeBrun ^[8]

References

1. 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.
2. 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.
3. Fujiki, Akira (1982). "On the douady space of a compact complex space in the category ${\displaystyle {\mathcal {C}}}$ ". Nagoya Mathematical Journal. 85: 189–211. doi:10.1017/S002776300001970X. MR 0759679.
4. 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.
5. 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
6. Fujiki, Akira (1983). "On a Compact Complex Manifold in ${\displaystyle {\mathcal {C}}}$  without Holomorphic 2-Forms". Publications of the Research Institute for Mathematical Sciences. 19: 193–202. doi:10.2977/PRIMS/1195182983. MR 0700948.
7. K. Ueno, ed., "Open Problems," Classification of Algebraic and Analytic Manifolds, Birkhaser, 1983.
8. Claude LeBrun, Yat-Sun Poon, "Twistors, Kahler manifolds, and bimeromorphic geometry II", J. Amer. Math. Soc. 5 (1992) MR1137099