# Is the Baily--Borel compactification functorial?

The following question seems pretty natural, but searching online and looking in some obvious places didn't turn up much, so maybe I can ask it here. (Disclaimer: I'm a newcomer to this topic, so apologies if the question is obviously misguided.)

Suppose that $V_1 = X_1/\Gamma_1$ and $V_2= X_2/\Gamma_2$ are arithmetic quotients of Hermitian symmetric domains. Let $V_1^\ast$ and $V_2^\ast$ be their respective Baily--Borel compactifications. Now suppose we have an analytic map $f: V_1 \rightarrow V_2$. Does it extend to a morphism $f^\ast: V_1^\ast \rightarrow V_2^\ast$?

If the answer in general is no, are there any nontrivial cases in which it is yes? (For instance, the simplest example that comes to mind is the case where $X_1=X_2$, $\Gamma_1 \subsetneq \Gamma_2$, and $f: V_1 \rightarrow V_2$ is the quotient.)

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A useful reference might be the article "Satake Compactification and extension of Holomorphic Mappings", Inv.Math. 16, 237-248, 1972, by Kiernan and Kobayashi. They show that if the map $V_1 \to V_2$ is induced from a map $X_1 \to X_2$ then it extends. In particular, the answer is positive for your "simplest example", though that presumably follows from the construction of the compactification.
Actually, I think the reference that you cite gives a positive answer to the original question (in most cases). See the remark after theorem 2 : a map $V_1\rightarrow V_2$ comes from a map $X_1\rightarrow X_2$ as soon as the action of $\Gamma_2$ on $X_2$ is free, but that is true as soon as $\Gamma_2$ is torsion-free (and this is an assumption that is very often made anyway). – Alex Feb 8 '11 at 15:03