---
tag: ZCBFF6DNA9
kind: definition
name: Moishezon manifold
msc: [32J10, 32J18]
defines: 8ZW9P9JRQD
uses: [JSGWAN2RJV, F0GF0JEZTG]
sources:
  - url: https://en.wikipedia.org/wiki/Moishezon_manifold
---

## Definition

Let $X$ be a compact, connected [[complex manifold|JSGWAN2RJV]] of dimension $n$, and let $a(X)$ be its [[algebraic dimension|F0GF0JEZTG]].
Then $X$ is a *Moishezon manifold* if $a(X) = n$, that is, if $X$ has $n$ algebraically independent meromorphic functions.

## Remarks

Example: $\mathbb{CP}^n$ is Moishezon, since $z_1/z_0, \ldots, z_n/z_0$ are $n$ algebraically independent meromorphic functions.
More generally, every smooth projective variety of dimension $n$ is Moishezon, since its field of rational functions has transcendence degree $n$.

Non-example: the diagonal [[Hopf surface|PNFNGBXRM3]] has dimension $2$ but algebraic dimension $1$, so it is not Moishezon.

Since $a(X) \le \dim X$ by [[Siegel's theorem|ENXCWPTG92]], Moishezon manifolds are those with the largest possible algebraic dimension.
Equivalently, $X$ is Moishezon if and only if it is bimeromorphic to a projective manifold.
A Moishezon manifold is projective if and only if it admits a Kähler metric, so non-projective Moishezon manifolds are never [[Kähler|X60JG9AEQJ]].