Let $X$ and $Y$ be complex manifolds with $\dim_{\mathbb C} X = m$ and $\dim_{\mathbb C} Y = n$. Let $F: X \to Y$ be a holomorphic map, and let $p \in X$. Suppose that $F$ is a holomorphic submersion at $p$, meaning that the complex-linear differential
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\begin{align*}
dF_p: T_p X \to T_{F(p)}Y
\end{align*}
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is surjective. Then $m \geq n$, and there exist holomorphic coordinate charts $(U,z)$ about $p$ and $(V,w)$ about $F(p)$, with $F(U) \subseteq V$, such that $z(p)=0$, $w(F(p))=0$, and
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\begin{align*}
(w \circ F \circ z^{-1})(z_1,\dots,z_m) = (z_1,\dots,z_n)
\end{align*}
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for all $(z_1,\dots,z_m)$ in the coordinate image $z(U) \subseteq \mathbb C^m$.