Let $X$ be a Hausdorff complex analytic space. Below, let $D$ be the open unit disc in $\mathbb{C}$. Let $D^\*$ be the punctured open unit disc.


I am looking for an analogue of the valuative criterion of properness in complex analysis.


Is the following correct?



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> > The complex analytic space $X$ is compact if every holomorphic map $D^\*\to X$ extends to a holomorphic map $D\to X$.
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The converse implication is not true, because there are non-extendable maps from $D^\*$ to $\mathbb{P}^1$, e.g., $z\mapsto \exp(-1/z^2)$.


I am thinking of $D^\*$ as Spec $K$ and $D $ as Spec $R$, where $R$ is a dvr with fraction field $K$.



What is the "analytic" analogue of the valuative criterion of properness
No, the open disk in $\mathbb C$ is a counterexample (removable singularity theorem plus maximum principle).


