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Suppose $F\colon\Omega\subseteq\mathbb C^m\to\mathbb C^n$ is a holomorphic map. WLOG, $0\in\Omega$ and $F(0)=0$. I want to determine the local normal form of $F$, i.e. classifying $F$ up to local biholomorphic coordinate changes: $$ \require{AMScd} \begin{CD} \Omega_0@>F>>\mathbb C_0^n\\ @V\sim VV@V\sim VV\\ \Omega_0'@>F'>>\mathbb C_0^n \end{CD} $$

When $m=n=1$, the result is well-known: $F$ is either a constant function, or locally $z\mapsto z^n$. On the other hand, when $F'(0)$ is injective or surjective, then implicit function theorem tells us that $F$ is locally equivalent to $F'(0)$, which is classified up to congruence. When $m,n>1$, maybe there's no complete classification, but I want some partial results.

Thoughts: Maybe we can view the problem formally. Instead of holomorphic maps, we can consider formal power series, especially when $n=1$. Weierstrass preparation theorem might work.

Background: I learnt that injective holomorphic maps $\mathbb C^n\to\mathbb C^n$ are biholomorphic. The proof in the course is approximately this. However, such a proof doesn't give any information on the classification of maps. I'm just looking for a result which is strong enough to prove that proposition.

Any idea? Thanks!

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  • $\begingroup$ Google for Weierstrass preparation theorem for a general local. I think Arnold worked on it too, for a classification in low dimensions, but I don't remember in which book he wrote it. Maybe in the volumes about dynamical systems!? $\endgroup$ – Nathanson Mar 28 '15 at 7:29

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