Several complex variables and analytic spaces Definition

Fujiki class C

Builds on Complex manifold Kähler manifold Modification of a complex space

Definition

Let $X$ be a compact, connected complex manifold. Then $X$ is in Fujiki class $\mathcal{C}$ if there are a compact Kähler manifold $Y$ and a compact complex manifold $Z$ with the following property. There are modifications $p \colon Z \to X$ and $q \colon Z \to Y$ . Here a surjective holomorphic map $f \colon Z \to V$ is a modification if it satisfies the following condition. There is a nowhere dense analytic subset $A \subset V$ such that $f^{-1}(A)$ is nowhere dense in $Z$ and $f$ maps $Z \setminus f^{-1}(A)$ biholomorphically onto $V \setminus A$ . In other words, $X$ is bimeromorphic to a compact Kähler manifold.

Remarks

Fujiki's original definition, for reduced compact complex spaces, asks that $X$ be the image of a compact Kähler space under a meromorphic map. By a theorem of Varouchas, for compact complex manifolds this is equivalent to the definition above.

Example: every compact Kähler manifold $X$ is in class $\mathcal{C}$ , with $Y = Z = X$ and $p = q = \id$ . Every Moishezon manifold is in class $\mathcal{C}$ , since it is bimeromorphic to a projective manifold, which is Kähler.

Non-example: a Hopf surface is not in class $\mathcal{C}$ . Manifolds in class $\mathcal{C}$ have even odd Betti numbers, as compact Kähler manifolds do, while a Hopf surface has $b_1 = 1$ .

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Tag	`WV1Q8N8YW7`
URI	`m2://WV1Q8N8YW7`
Kind	definition
Writing	W0 stub
MSC	32J27
Defines	Fujiki class C
Uses	Complex manifold, Kähler manifold, Modification of a complex space

Fujiki class C

Definition

Remarks

Reading path

Further reading

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