TITLE: What is the "analytic" analogue of the valuative criterion of properness
QUESTION [10 upvotes]: 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?


The complex analytic space $X$ is compact if every holomorphic map $D^*\to X$ extends to a holomorphic map $D\to X$.


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 $K$.

REPLY [11 votes]: No, the open disk in $\mathbb C$ is a counterexample (removable singularity theorem plus maximum principle).