---
tag: M28SB04806
kind: definition
name: Envelope of holomorphy
msc: [32D10, 32D26]
defines: X71JE6A3ZJ
uses: [A15BF6QHDR, 507C34W0ZT, 3MQTWKBBBT]
sources:
  - url: https://en.wikipedia.org/wiki/Domain_of_holomorphy
---

## Definition

Let $D \subseteq \C^n$ be a domain, that is, a nonempty connected open set.
A *holomorphic extension* of $D$ is a triple $(X, \pi, \alpha)$ with two properties.

1. $(X, \pi)$ is a [[Riemann domain over $\C^n$|A15BF6QHDR]] and $\alpha \colon D \to X$ is a [[holomorphic map|3MQTWKBBBT]] with $\pi \circ \alpha = \id_D$.
2. For every [[holomorphic function|507C34W0ZT]] $f$ on $D$ there is a holomorphic function $F$ on $X$ with $F \circ \alpha = f$.

An *envelope of holomorphy* of $D$ is a holomorphic extension $(E, \pi, \alpha)$ of $D$ with the following property.
For every holomorphic extension $(X, \rho, \beta)$ of $D$ there is a holomorphic map $\tau \colon X \to E$ with $\pi \circ \tau = \rho$ and $\tau \circ \beta = \alpha$.

## Remarks

The function $F$ in clause 2 is unique, because $X$ is connected and $\alpha(D)$ is a nonempty open subset of $X$.
Every domain $D$ has an envelope of holomorphy, and it is unique up to an isomorphism of Riemann domains compatible with the maps from $D$.
It can be built from the germs of holomorphic functions obtained from $D$ by analytic continuation.

Example: for the [[Hartogs figure|GHJC53RFBT]] $H$ in $\C^n$, the polydisc $\Delta^n$ with the inclusion map is an envelope of holomorphy.
Every holomorphic function on $H$ extends to $\Delta^n$; and $\Delta^n$ is a domain of holomorphy, which gives the maximality.

Non-example: for $D = \set{z \in \C^2 : 0 < \abs{z} < 1}$, the triple $(D, \id, \id)$ is a holomorphic extension but not an envelope.
The extension given by the ball $\set{\abs{z} < 1}$ admits no map $\tau$ into $D$ over $\C^2$, since $\tau$ would have to be the identity on the ball.

A domain $D$ is a domain of holomorphy exactly when $(D, \id, \id)$ is its envelope of holomorphy.