Reference for the analytic valuative criterion of properness

Solution 1:

I don't know a reference, but here's a probable (I haven't double checked all the details) proof that's probably way overkill. We can assume throughout that $X$ is irreducible, as I leave it to you to check.

If $X$ is one-dimensional, this is easy. Indeed, we just directly verify that $X$ satisfies the existence part of the valuative criterion for properness (the uniqueness part follows from the assumption of separatedness). Indeed, let $\overline{X}$ be a projective closure of $X$ (note that $X$ is automatically quasi-projective--see this--so that this projective closure makes sense). Map a small disk $\Delta$ to $\overline{X}^\mathrm{an}$ such that $0\mapsto p\in \overline{X}-X$ for which $\Delta^\ast$ (the punctured disk) maps to $X$ (take a very small disk around any preimage point of $p$ in a normalization of $\overline{X}$) and consider the tautological map $\Delta^\ast\to X^\mathrm{an}$. This obviously cannot be extended to a map $\Delta\to X$ because by continuity the composition $\Delta\to \overline{X}^\mathrm{an}\to X$ must send $0$ to $p$, which is not in $X^\mathrm{an}$.

In general, let $\overline{X}$ be a compactification of $X$ which contains $X$ as a dense open (see this and this)--we may again assume WLOG that $\overline{X}$ is irreducible. Let $p$ be a point of $\overline{X}-X$ and $q$ a point of $X$, and let $C$ be a normal curve connecting $p$ and $q$ (e.g. see Corollary 1.9 of this and take normalization). Then, by the same trick as in the one-dimensional case build a map $\Delta\to C$ so that $0\mapsto p$ and $\Delta^\ast$ maps into $C\cap X$ (note that this is a dense open). The same argument shows that you can't extend the resulting map $\Delta^\times\to X^\mathrm{an}$ to a map $\Delta\to X^\mathrm{an}$.

PS, it's false that this is necessary as there are maps $\Delta^\ast\to \mathbb{P}^{1,\mathrm{an}}_\mathbb{C}$ that can't be extended (e.g. $z\mapsto \exp(-z^{-2})$). In fact, I don't know an example of an $X$ satisfying this, and a wise elder has suggested that perhaps there aren't any (e.g. $X$ essentially has to be a point). This would also corroborate why one cannot find this result in books.