---
tag: 8ZW9P9JRQD
kind: concept
name: Moishezon manifold
msc: [32J10, 32J18]
specializes: [JSGWAN2RJV]
sources:
  - url: https://en.wikipedia.org/wiki/Moishezon_manifold
same_as:
  wikidata: Q17097705
  wikipedia: Moishezon manifold
---

## Summary

A Moishezon manifold is a compact connected [[complex manifold|JSGWAN2RJV]] with as many independent meromorphic functions as its dimension.
In other words, its [[algebraic dimension|F0GF0JEZTG]] equals its dimension.
Every projective manifold is Moishezon.
Moishezon proved that every Moishezon manifold becomes projective after finitely many blow-ups along smooth centres.
He also proved that a Moishezon manifold is projective if and only if it is [[Kähler|X60JG9AEQJ]].
So Moishezon manifolds are the compact complex manifolds closest to algebraic varieties; they are the manifolds that correspond to proper algebraic spaces.

## Explanations

### L1 human

A Moishezon manifold carries enough meromorphic functions to serve as coordinates almost everywhere.
Projective varieties, the spaces cut out by polynomial equations in projective space, are the main examples.
There are also Moishezon manifolds that are not projective, first found by Hironaka in dimension $3$, but they differ from projective ones only by a birational change.
In dimension $2$ every Moishezon manifold is projective.