Fujiki class C

In algebraic geometry, a complex manifold is called Fujiki class $\mathcal{C}$ if it is bimeromorphic to a compact Kähler manifold. This notion was defined by Akira Fujiki.{{cite journal |url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00209391X |title=On Automorphism Groups of Compact Kähler Manifolds |journal=Inventiones Mathematicae |year=1978 |volume=44 |pages=225–258 |last1=Fujiki |first1=Akira |issue=3 |doi=10.1007/BF01403162 |bibcode=1978InMat..44..225F|mr = 481142}}

Properties

Let M be a compact manifold of Fujiki class $\mathcal{C}$ , and

$X\subset M$ its complex subvariety. Then X

is also in Fujiki class $\mathcal{C}$ (,{{cite journal |doi=10.2977/PRIMS/1195189279 |title=Closedness of the Douady spaces of compact Kähler spaces |year=1978 |last1=Fujiki |first1=Akira |journal=Publications of the Research Institute for Mathematical Sciences |volume=14 |pages=1–52 | mr = 486648|doi-access=free }} 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 $\mathcal{C}$ .{{cite journal |doi=10.1017/S002776300001970X |title=On the douady space of a compact complex space in the category $\mathcal{C}$ . |year=1982 |last1=Fujiki |first1=Akira |journal=Nagoya Mathematical Journal |volume=85 |pages=189–211| mr = 759679|doi-access=free }}

Fujiki class $\mathcal{C}$ manifolds are examples of compact complex manifolds which are not necessarily Kähler, but for which the $\partial \bar \partial$ -lemma holds.{{cite journal |doi=10.1007/s00222-012-0406-3 |title=On the $\partial \bar \partial$ -Lemma and Bott-Chern cohomology |year=2013 |last1=Angella |first1=Daniele |last2=Tomassini |first2=Adriano |journal=Inventiones Mathematicae |volume=192 |pages=71–81 |s2cid=253747048 |url=http://eprints.adm.unipi.it/1612/1/angella%2Dtomassini%2Drevised.pdf }}

Conjectures

J.-P. Demailly and M. Pǎun have

shown that a manifold is in Fujiki class $\mathcal{C}$ if and only

if it supports a Kähler current.Demailly, Jean-Pierre; Pǎun, Mihai [https://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 = 2113021}}

They also conjectured that a manifold M is in Fujiki class $\mathcal{C}$ if it admits a nef current which is big, that is, satisfies

: $\int_M \omega^{{dim_{\mathbb C} M}}>0.$

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

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

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

Fujiki{{cite journal |doi=10.2977/PRIMS/1195182983 |title=On a Compact Complex Manifold in $\mathcal{C}$ without Holomorphic 2-Forms |year=1983 |last1=Fujiki |first1=Akira |journal=Publications of the Research Institute for Mathematical Sciences |volume=19 |pages=193–202|mr = 700948|doi-access=free }} and UenoK. Ueno, ed., "Open Problems," Classification of Algebraic and Analytic Manifolds, Birkhaser, 1983. asked whether the property $\mathcal{C}$ is stable under deformations. This conjecture was disproven in 1992 by Y.-S. Poon and Claude LeBrun Claude LeBrun, Yat-Sun Poon, [https://arxiv.org/abs/alg-geom/9202006 "Twistors, Kahler manifolds, and bimeromorphic geometry II"], J. Amer. Math. Soc. 5 (1992) {{MathSciNet | id = 1137099}}

References

Category:Algebraic geometry

Category:Complex manifolds