Let $X$ and $Y$ be complex manifolds of complex dimensions $m$ and $n$, respectively, with $m,n \in \mathbb{N}$. Let $F: X \to Y$ be a holomorphic map, and let $p \in X$. Suppose that $F$ is a holomorphic immersion at $p$, meaning that the complex-linear differential
is injective. Then $m \leq n$, and there exist open neighborhoods $U \subset X$ of $p$ and $V \subset Y$ of $F(p)$, with $F(U) \subseteq V$, together with holomorphic coordinate charts
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\begin{align*}
z: U \to z(U) \subseteq \mathbb{C}^m
\end{align*}
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and
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\begin{align*}
w: V \to w(V) \subseteq \mathbb{C}^n
\end{align*}
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such that $z(p)=0$, $w(F(p))=0$, and, for every $\zeta=(\zeta_1,\dots,\zeta_m) \in z(U)$,
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\begin{align*}
w \circ F \circ z^{-1}(\zeta_1,\dots,\zeta_m) = (\zeta_1,\dots,\zeta_m,0,\dots,0),
\end{align*}
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where the final $0$ entries form the zero vector in $\mathbb{C}^{n-m}$.