---
tag: 282MQP4X82
kind: statement
role: theorem
name: Holomorphic inverse function theorem
msc: [32H02, 32A10]
uses: [3MQTWKBBBT, FX92PP3HZR, 0ZCSMYHTBP, 0K9HSJFCMJ]
about: [3MQTWKBBBT, FX92PP3HZR]
sources:
  - url: https://en.wikipedia.org/wiki/Inverse_function_theorem
---

## Statement

Let $U \subseteq \C^n$ be an open set, let $F = (F_1, \dots, F_n) \colon U \to \C^n$ be a [[holomorphic map|3MQTWKBBBT]], and let $a \in U$.
Suppose that the complex [[Jacobian determinant|0ZCSMYHTBP]] $\det \left( \frac{\partial F_k}{\partial z_j}(a) \right)_{k, j}$ is not zero.
Then there are open neighbourhoods $U_0 \subseteq U$ of $a$ and $V_0$ of $F(a)$ such that $F$ restricts to a [[biholomorphism|FX92PP3HZR]] from $U_0$ onto $V_0$.

## Remarks

Viewed as a map of $\R^{2n}$, $F$ has real Jacobian determinant $\abs{\det (\partial F_k / \partial z_j)}^2$, which is not zero at $a$.
The real [[inverse function theorem|0K9HSJFCMJ]] gives a $C^1$ local inverse $G$.
Its derivative at each point is the inverse of the $\C$-linear map $DF$, hence $\C$-linear, so $G$ is holomorphic.
Example: $F(z_1, z_2) = (e^{z_1}, \ z_2 e^{z_1})$ has complex Jacobian determinant $e^{2 z_1} \ne 0$ everywhere, so it is a local biholomorphism near every point, though it is not injective on $\C^2$, since $F(z_1 + 2\pi i, z_2) = F(z_1, z_2)$.
The hypothesis is needed: $F(z) = z^2$ on $\C$ has derivative $0$ at $0$, and it is not injective on any neighbourhood of $0$.

## Explanations

### L1 human

If the complex derivative of a holomorphic map is invertible at a point, the map can be undone near that point.
Moreover, the inverse is again holomorphic.
So holomorphic maps with invertible derivative are local changes of complex coordinates.

### L2 human

This theorem is the basic local tool for complex manifolds and holomorphic maps.
Together with the holomorphic implicit function theorem, it shows that the zero set of holomorphic functions with independent differentials is locally a complex submanifold.
A stronger fact holds in the complex case: an injective holomorphic map between open subsets of $\C^n$ automatically has nonzero Jacobian determinant, so it is a biholomorphism onto its image.
The real analogue fails, as $x \mapsto x^3$ shows.