[proofplan] Choose arbitrary holomorphic coordinates near $p$ and $F(p)$, and write $F$ locally as a holomorphic map between open subsets of complex Euclidean spaces. Since the differential at $p$ has rank $m$, some $m \times m$ target-coordinate minor is invertible; after reordering target coordinates, the first $m$ component functions form a locally biholomorphic map. Use those component functions as the new source coordinates, so the map becomes the graph of a [holomorphic function](/page/Holomorphic%20Function) $h: \mathbb{C}^m \to \mathbb{C}^{n-m}$. Finally subtract this graph in the target coordinates to flatten the image to the standard coordinate inclusion. [/proofplan]

[step:Write the map in local coordinates and choose an invertible Jacobian minor]Choose holomorphic coordinate charts $(U_0,\varphi)$ around $p$ and $(V_0,\psi)$ around $F(p)$ such that $F(U_0) \subseteq V_0$. By translating the coordinate values, assume $\varphi(p)=0 \in \mathbb{C}^m$ and $\psi(F(p))=0 \in \mathbb{C}^n$. Define the local representative \begin{align*} f: \varphi(U_0) \to \psi(V_0), \qquad f = \psi \circ F \circ \varphi^{-1}. \end{align*} This is a holomorphic map between open subsets of $\mathbb{C}^m$ and $\mathbb{C}^n$, and its complex differential at $0$ represents $dF_p$ under the coordinate identifications \begin{align*} d\varphi_p: T_pX \to \mathbb{C}^m \end{align*} and \begin{align*} d\psi_{F(p)}: T_{F(p)}Y \to \mathbb{C}^n. \end{align*} Since $dF_p$ is injective, the complex-[linear map](/page/Linear%20Map) \begin{align*} df_0: \mathbb{C}^m \to \mathbb{C}^n \end{align*} has rank $m$. Hence $m \leq n$, and the $n \times m$ Jacobian matrix $Jf_0$ has an invertible $m \times m$ row minor. After permuting the target coordinates in $\mathbb{C}^n$, assume this minor is formed by the first $m$ component functions of $f$. Let \begin{align*} \pi: \mathbb{C}^n \to \mathbb{C}^m, \qquad \pi(\xi_1,\dots,\xi_n) = (\xi_1,\dots,\xi_m) \end{align*} be the projection onto the first $m$ coordinates, and define \begin{align*} g: \varphi(U_0) \to \mathbb{C}^m, \qquad g = \pi \circ f. \end{align*} Then $g$ is holomorphic and its Jacobian matrix $Jg_0$ is the selected invertible $m \times m$ minor of $Jf_0$.[/step]

[guided]We first reduce the statement to a theorem about holomorphic maps between open subsets of complex Euclidean spaces. Choose holomorphic coordinate charts $(U_0,\varphi)$ near $p$ and $(V_0,\psi)$ near $F(p)$, shrinking $U_0$ if necessary so that $F(U_0) \subseteq V_0$. Translating the coordinate maps by constants does not change their holomorphic coordinate property, so we may arrange \begin{align*} \varphi(p)=0 \in \mathbb{C}^m \end{align*} and \begin{align*} \psi(F(p))=0 \in \mathbb{C}^n. \end{align*} Define the coordinate expression of $F$ by \begin{align*} f: \varphi(U_0) \to \psi(V_0), \qquad f = \psi \circ F \circ \varphi^{-1}. \end{align*} The map $f$ is holomorphic because it is a composition of holomorphic maps. The coordinate identifications of tangent spaces are the complex-linear isomorphisms \begin{align*} d\varphi_p: T_pX \to \mathbb{C}^m \end{align*} and \begin{align*} d\psi_{F(p)}: T_{F(p)}Y \to \mathbb{C}^n. \end{align*} Under these identifications, the differential $dF_p$ is represented by \begin{align*} df_0: \mathbb{C}^m \to \mathbb{C}^n. \end{align*} Because $F$ is a holomorphic immersion at $p$, the map $dF_p$ is injective. Therefore $df_0$ is injective as a complex-linear map from an $m$-dimensional complex [vector space](/page/Vector%20Space) into an $n$-dimensional complex vector space. This implies $m \leq n$ and $\operatorname{rank} df_0 = m$. Equivalently, the Jacobian matrix $Jf_0$ has rank $m$, so some $m \times m$ row minor is invertible. The rows of $Jf_0$ correspond to target coordinates. A permutation of the target coordinates is a complex-linear biholomorphic change of coordinates on $\mathbb{C}^n$, so after applying such a permutation we may assume that the invertible minor uses the first $m$ target components. Let \begin{align*} \pi: \mathbb{C}^n \to \mathbb{C}^m, \qquad \pi(\xi_1,\dots,\xi_n) = (\xi_1,\dots,\xi_m) \end{align*} be the projection onto those first $m$ components, and set \begin{align*} g: \varphi(U_0) \to \mathbb{C}^m, \qquad g = \pi \circ f. \end{align*} Then $g$ is holomorphic, and $Jg_0$ is exactly the selected invertible minor. This is the point of choosing the minor: it gives a square holomorphic map whose derivative is invertible, so that the [holomorphic inverse function theorem](/theorems/4950) can be applied in the next step.[/guided]

[step:Use the first target components as source coordinates] Since $\varphi(U_0)$ is an open subset of $\mathbb{C}^m$, $g: \varphi(U_0) \to \mathbb{C}^m$ is holomorphic, and $Jg_0$ is invertible, the Holomorphic [Inverse Function Theorem](/theorems/51) applies to $g$ at $0$. Therefore there exist open neighborhoods $\Omega_1 \subseteq \varphi(U_0)$ of $0$ and $\Omega_2 \subseteq \mathbb{C}^m$ of $0$ such that \begin{align*} g|_{\Omega_1}: \Omega_1 \to \Omega_2 \end{align*} is biholomorphic. Let \begin{align*} G: \Omega_2 \to \Omega_1 \end{align*} denote the holomorphic inverse of $g|_{\Omega_1}$. Shrink $U_0$ to $U := \varphi^{-1}(\Omega_1)$. Define the source coordinate map \begin{align*} z: U \to \Omega_2, \qquad z = g \circ \varphi. \end{align*} Then $z$ is a holomorphic coordinate chart around $p$, and $z(p)=0$. [/step]

[step:Express the map as a holomorphic graph] Let \begin{align*} \rho: \mathbb{C}^n \to \mathbb{C}^{n-m}, \qquad \rho(\xi_1,\dots,\xi_n) = (\xi_{m+1},\dots,\xi_n) \end{align*} be the projection onto the last $n-m$ coordinates. Define \begin{align*} h: \Omega_2 \to \mathbb{C}^{n-m}, \qquad h = \rho \circ f \circ G. \end{align*} The map $h$ is holomorphic as a composition of holomorphic maps. For every $u \in \Omega_2$, \begin{align*} f(G(u)) = (u,h(u)). \end{align*} Indeed, the first $m$ coordinates of $f(G(u))$ are \begin{align*} \pi(f(G(u))) = g(G(u)) = u, \end{align*} while the last $n-m$ coordinates are, by definition, \begin{align*} \rho(f(G(u))) = h(u). \end{align*} Thus, in the source coordinate $u=z(x)$, the map $F$ is locally the graph of $h$. [/step]

[step:Flatten the graph by changing target coordinates] Choose an open neighborhood $\Lambda_1 \subseteq \psi(V_0) \cap (\Omega_2 \times \mathbb{C}^{n-m})$ of $f(\Omega_1)$, so that the first $m$ coordinates of every point of $\Lambda_1$ lie in the domain $\Omega_2$ of $h$. Define \begin{align*} \Theta: \Lambda_1 \to \mathbb{C}^m \times \mathbb{C}^{n-m}, \qquad \Theta(u,v) = (u, v - h(u)), \end{align*} where $u \in \mathbb{C}^m$ denotes the first $m$ coordinates and $v \in \mathbb{C}^{n-m}$ denotes the last $n-m$ coordinates. Shrinking $\Lambda_1$ if necessary, the inverse map is \begin{align*} \Theta^{-1}: \Theta(\Lambda_1) \to \Lambda_1, \qquad \Theta^{-1}(u,\eta) = (u,\eta + h(u)). \end{align*} Both $\Theta$ and $\Theta^{-1}$ are holomorphic, so $\Theta$ is a biholomorphic coordinate change on the target. Shrink $U$ further, if necessary, so that $F(U) \subseteq V := \psi^{-1}(\Lambda_1)$. Define the target coordinate map \begin{align*} w: V \to \Theta(\Lambda_1), \qquad w = \Theta \circ \psi. \end{align*} Then $w$ is a holomorphic coordinate chart around $F(p)$. Since $g(0)=\pi(f(0))=0$ and $G$ is the inverse of $g|_{\Omega_1}$, we have $G(0)=0$. Hence $h(0)=\rho(f(G(0)))=\rho(f(0))=0$, and therefore $w(F(p))=\Theta(f(0))=(0,0)$. For $x \in U$, write $u=z(x)$. Since $z=g \circ \varphi$, we have $\varphi(x)=G(u)$. Therefore \begin{align*} w(F(x)) = \Theta(\psi(F(x))) = \Theta(f(\varphi(x))) = \Theta(f(G(u))). \end{align*} Using the graph expression $f(G(u))=(u,h(u))$, this becomes \begin{align*} w(F(x)) = \Theta(u,h(u)) = (u,h(u)-h(u)) = (u,0). \end{align*} Since $u=z(x)$, this is precisely \begin{align*} w \circ F \circ z^{-1}(u) = (u,0) \end{align*} for all $u \in z(U)$. Written in coordinates, \begin{align*} w \circ F \circ z^{-1}(\zeta_1,\dots,\zeta_m) = (\zeta_1,\dots,\zeta_m,0,\dots,0). \end{align*} This is the asserted local normal form. [/step]