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I have originally posted this question on math.SE, but it received no attention, so I repost it here.

Let $U$ be an open domain in $\mathbb{C}^{n}$. Let $m\ge n$ and let $F:U\to C^{m}$ be a holomorphic map.

What are the sufficient conditions for $F$ to have a discrete set of critical points?

One can show that if $F$ is an injection, then the set of critical points is thin, and so if $n=1$, this is sufficient. I am interested in a general case.

The set of critical points is an intersection of zero-sets of the Jacobians of projections of $F$, so I hope that there is some geometric condition on $F$, which would imply, that these submanifolds are positioned in the appropriate "non-tangential" way.

Thank you.

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  • $\begingroup$ Maybe this is more simplistic than what you're looking for: there is a canonical holomorphic map $\Lambda^n(DF):U\to \operatorname{Hom}(\Lambda^n\mathbb{C}^n,\Lambda^n\mathbb{C}^m)$, given by sending $z\in U$ to the linear map $\Lambda^n(DF|_z):\Lambda^n\mathbb{C}^n\to\Lambda^n\mathbb{C}^m$. (Note that $\operatorname{Hom}(\Lambda^n\mathbb{C}^n,\Lambda^n\mathbb{C}^m)$ is a complex vector space of dimension ${m \choose n}$.) The critical points of $F$ are isolated if and only if the zeroes of $\Lambda^n(DF)$ are isolated. $\endgroup$
    – macbeth
    Commented Oct 16, 2017 at 8:13
  • $\begingroup$ @macbeth but how to deal with that map? $\endgroup$
    – erz
    Commented Oct 18, 2017 at 3:42

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