---
tag: F0GF0JEZTG
kind: concept
name: Algebraic dimension
msc: [32J10, 32J18]
sources:
  - url: https://en.wikipedia.org/wiki/Moishezon_manifold
---

## Summary

The algebraic dimension $a(X)$ of a compact connected [[complex manifold|JSGWAN2RJV]] $X$ is the [[transcendence degree|D7TR0YWR9H]] over $\C$ of its field of meromorphic functions.
It measures how many independent meromorphic functions $X$ carries.
By [[Siegel's theorem|ENXCWPTG92]], $0 \le a(X) \le \dim X$.
The extreme case $a(X) = \dim X$ defines [[Moishezon manifolds|8ZW9P9JRQD]], which are close to projective varieties, while $a(X) = 0$ occurs for generic complex tori.

## Explanations

### L1 human

On a compact complex manifold the holomorphic functions are all constant, but meromorphic functions, which may have poles, can be interesting.
The algebraic dimension counts how many of them are independent, in the sense that no polynomial relation links them.
On complex projective space of dimension $n$ there are $n$ independent ones, such as $z_1/z_0, \ldots, z_n/z_0$.
On some complex tori there are none besides the constants.
So the algebraic dimension tells how far a compact complex manifold is from being algebraic.