---
tag: H90N8NBVNY
kind: concept
name: Fujiki class C
msc: [32J27]
sources:
  - url: https://en.wikipedia.org/wiki/Fujiki_class_C
same_as:
  wikidata: Q5507425
  wikipedia: Fujiki class C
---

## Summary

Fujiki class $\mathcal{C}$ is the class of compact complex manifolds that are bimeromorphic to compact [[Kähler manifolds|X60JG9AEQJ]].
It contains all compact Kähler manifolds and all [[Moishezon manifolds|8ZW9P9JRQD]].
Unlike the Kähler property, membership in class $\mathcal{C}$ is invariant under bimeromorphic maps and under images of holomorphic maps.
Manifolds in class $\mathcal{C}$ keep much of the topology of Kähler manifolds: their cohomology has a Hodge decomposition, and their odd Betti numbers are even.

## Explanations

### L1 human

Kähler manifolds are compact complex manifolds with a particularly nice kind of metric.
Blowing up or down, or other birational changes, can destroy that metric, even though the manifold stays almost the same.
Class $\mathcal{C}$ is the bigger class one gets by allowing such changes.
It is the natural home for results about Kähler manifolds that only depend on the manifold up to birational change.