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I have seen in several places that, given a holomorphic map $F:M^m \rightarrow N^n$ between complex manifolds, the level sets of $F$ support(/are) subvarieties. That is, for any $y \in N$ the set $F^{-1}(y) \subset M$ is a subvariety of $M$.

I would like to know how to prove this precisely, in fact I would like to know as many different ways to prove this as possible.

There are a few questions along these lines on this site but I wanted a clarification on this.

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    $\begingroup$ If you work in local coordinates, you have a map $F=(F_1,\dots, F_n):\mathbb C^m\to\mathbb C^n$ and you can assume that your coordinates are chosen such that $F(0)=0$. Now, $F^{-1}(0)=\{z\mid F(z)=0\}$ is what is called an analytic set. Since $F$ is holomorphic, the analytic sets defined this way will most likely give an analytic set on $M$. You should be aware that this analytic set may define a singular object. Consider for example the map $f: \mathbb C^n\to \mathbb C, (z_1,\dots,z_n)\mapsto z_1^2+\dots+z_n^2$. $\endgroup$ – James Sep 27 '18 at 13:24
  • $\begingroup$ Thank you! Do you know if it is possible to prove this without recourse to local (holomorphic) coordinates? $\endgroup$ – ben Sep 28 '18 at 7:55
  • $\begingroup$ Thank you for the edit Bernard. $\endgroup$ – ben Oct 2 '18 at 8:36

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