---
tag: WV1Q8N8YW7
kind: definition
name: Fujiki class C
msc: [32J27]
defines: H90N8NBVNY
uses: [JSGWAN2RJV, X60JG9AEQJ, SB3MY7XQM9]
sources:
  - url: https://en.wikipedia.org/wiki/Fujiki_class_C
---

## Definition

Let $X$ be a compact, connected [[complex manifold|JSGWAN2RJV]].
Then $X$ is *in Fujiki class $\mathcal{C}$* if there are a compact [[Kähler manifold|X60JG9AEQJ]] $Y$ and a compact complex manifold $Z$ with the following property.
There are [[modifications|SB3MY7XQM9]] $p \colon Z \to X$ 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|8ZW9P9JRQD]] is in class $\mathcal{C}$, since it is bimeromorphic to a projective manifold, which is Kähler.

Non-example: a [[Hopf surface|PNFNGBXRM3]] 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 $b_1 = 1$.