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The holomorphic inverse function theorem in several complex variables

Statement

Let $m \geq 1$ , let $U \subseteq C^{m}$ be open, let $F : U \to C^{m}$ be holomorphic, and let $a \in U$ . If $\det J_{C} F (a) \neq 0$ , then there are open neighbourhoods $U_{0} \subseteq U$ of $a$ and $V_{0}$ of $F (a)$ such that $F ∣_{U_{0}} : U_{0} \to V_{0}$ is biholomorphic.

If $G : V_{0} \to U_{0}$ denotes the inverse, then

$D G (w) = D F (G (w))^{- 1} (w \in V_{0}) .$

Facts & Assumptions

Given: The open set $U$ , the holomorphic map $F : U \to C^{m}$ , and the point $a \in U$ with $\det J_{C} F (a) \neq 0$ .

[L1]

A holomorphic map into $C^{m}$ has holomorphic scalar components, and holomorphic scalar functions of several variables are smooth in the real coordinates (A map into $C^{n}$ is holomorphic exactly when each of its components is, Holomorphic functions of several variables are smooth and their complex derivatives are holomorphic).

[L2]

For a complex-linear automorphism, the real Jacobian determinant is the squared modulus of the complex determinant (The real Jacobian determinant of a complex-linear automorphism is the squared modulus of its complex determinant).

[L3]

A $C^{1}$ map of open subsets of $R^{2 m}$ with invertible real derivative has a local $C^{1}$ inverse, and that inverse derivative is the inverse linear map (The Euclidean inverse function theorem).

[F1]

A biholomorphism is a bijective holomorphic map with holomorphic inverse (Biholomorphic maps between open sets in $C^{m}$ ).

Proof

technique · direct

1.1givenL1

By [L1], every component of $F$ is holomorphic and therefore smooth as a real-valued pair of functions on $R^{2 m}$ . Hence the underlying real map $F : U \subseteq R^{2 m} \to R^{2 m}$ is $C^{1}$ on a neighbourhood of $a$ .

1.2givenL2

The real derivative $D F (a)$ is the same linear map as the complex differential, now read over $R$ . Since $\det J_{C} F (a) \neq 0$ , [L2] gives $\det_{R} D F (a) \neq 0$ , so $D F (a)$ is invertible as a real linear map.

2.1step 1.1step 1.2L3construct

Apply [L3] to the real map from step 1.1 and the invertible derivative from step 1.2. This gives open neighbourhoods $U_{0}$ of $a$ and $V_{0}$ of $F (a)$ such that $F ∣_{U_{0}} : U_{0} \to V_{0}$ is bijective and has a $C^{1}$ inverse $G : V_{0} \to U_{0}$ . Moreover, $D G (w) = D F (G (w))^{- 1} (w \in V_{0}) .$

3.1step 2.1F1algebra∎

For each $w \in V_{0}$ , the linear map $D F (G (w))$ is $C$ -linear because $F$ is holomorphic, so its inverse $D G (w)$ is also $C$ -linear. Step 2.1 already gives $G$ real differentiable at every point with that differential, hence the defining linear approximation for holomorphy uses a $C$ -linear derivative. Therefore $G$ is holomorphic on $V_{0}$ . Together with the holomorphy of $F ∣_{U_{0}}$ , [F1] makes $F ∣_{U_{0}}$ biholomorphic.

The holomorphic inverse function theorem in several complex variables

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources