---
tag: 2SM64J81YC
kind: statement
role: proposition
name: Holomorphic functions on compact complex manifolds are constant
msc: [32Q99, 32A10]
uses: [JSGWAN2RJV, 0TVJZYB6SH, 0V032VWTHD]
about: [JSGWAN2RJV]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_manifold
---

## Statement

Let $X$ be a [[compact|0TVJZYB6SH]], [[connected|0V032VWTHD]] [[complex manifold|JSGWAN2RJV]], and let $f \colon X \to \C$ be a holomorphic function.
Then $f$ is constant.

## Remarks

Proof idea: since $X$ is compact, $\abs{f}$ attains its maximum at some point $p$.
In a connected chart around $p$, the [[maximum modulus principle in several variables|91DW8V0PET]] shows that $f$ equals $f(p)$ near $p$.
The set of points near which $f \equiv f(p)$ is open, and by the [[identity theorem|7GZ908QZ04]] it is also closed, so by connectedness it is all of $X$.

Both hypotheses are needed.
On the non-compact manifold $\C^n$, the coordinate function $z_1$ is holomorphic and not constant.
On a disjoint union of two compact complex manifolds, take the function equal to $0$ on one piece and $1$ on the other.
It is holomorphic and not constant; in general $f$ is only locally constant.

Consequence: a compact connected complex manifold of positive dimension is not a complex submanifold of any $\C^N$, since the coordinate functions restricted to it would be constant.
For example, [[complex projective space|7GJ4STCNNM]] does not embed holomorphically in $\C^N$.

## Explanations

### L1 human

On a compact complex manifold, such as the Riemann sphere or a torus, every holomorphic function is constant.
The reason is the maximum principle: the absolute value of a holomorphic function cannot have a peak inside its domain.
On a compact space there is no boundary to escape to.
This is a strong contrast with smooth functions, which exist in great supply on every manifold.

### L2 human

This result is the several-variable version of the fact that a compact Riemann surface has no nonconstant holomorphic functions, and it is a cousin of [[Liouville's theorem|0EHKVRF4J6]].
Because global holomorphic functions are useless on a compact complex manifold, one studies instead meromorphic functions and holomorphic sections of line bundles.
The opposite extreme is a [[Stein manifold|HMCD0JBFST]], which has enough holomorphic functions to embed in some $\C^N$.