# Is the analytification of the coarse space equal to the coarse moduli space of the analytification?

If $X$ is a smooth finite type separated DM algebraic stack over $\mathbb C$ with coarse space $X^c$, then do we know whether the analytification of $X^c$ is the coarse space of the analytification of $X$? That is, do we have $$(X^c)^{an} = (X^{an})^c?$$

My stack X is actually even simpler, as its inertia stack I_X->X is finite etale. The coarse space $X^c$ of X is in this case the rigidification of X with respect to I_X, so that $X$ is an $I_X$-gerbe over $X^c$. I presume that the coarse space of $X^{an}$ should be the rigidification of $X^{an}$ with respect to $I_{X^{an}}$ in this case. Is that true?

"Definition." The coarse space of $X^{an}$ is a morphism of complex analytic stacks $X^{an} \to (X^{an})^c$ satisfying the usual universal properties, e.g., for a complex analytic space $V$ and morphism $X^{an}\to V$ there is a unique $(X^{an})^c\to V$ such that $X^{an}\to V$ factors via this morphism.

Idea: There is (by the universal property of $(X^{an})^c$) a morphism $(X^{an})^c\to (X^c)^{an}$. This morphism is finite of degree one, I think. Thus an application of "Zariski's Main Theorem" (in the analytic category) should conclude the proof. Does this work?

• What does "finite of degree 1" mean? That is, how do you define a useful notion of "degree" without knowing flatness? – nfdc23 Feb 18 '17 at 20:45
• @nfdc23 I see. Let's avoid using "degrees"...Is it not clear that this morphism is birational, i.e., an isomorphism over a dense open? I presume that even for (nice) complex analytic spaces a birational quasi-finite proper morphism should be an isomorphism. No? – George Feb 19 '17 at 16:26
• Yes, a proper quasi-finite morphism is finite (in the sense of coherent sheaves of algebras) in the analytic setting, so under normality hypotheses a bijective finite surjective map between dense analytic Zariski-open sets is an isomorphism. – nfdc23 Feb 19 '17 at 17:53
• @nfdc23 Ok. That's good to know. What's bothering me is the existence of the coarse space in the analytic category. Is there a Keel-Mori type of theorem for separated Deligne-Mumford complex analytic stacks? – George Feb 19 '17 at 18:36
• Have you tried taking some proof of the Keel-Mori theorem and see how it adapts to the analytic setting for such separated stacks (in effect, that the analytifications of the various algebraic steps have the required properties in the analytic category)? Assuming you have some familiarity with the basics of Stein spaces and complex-analytic spaces, you should try that and see how it goes. For example, by consideration of completed local rings one sees that forming naive quotient of an affine by the action of a finite group commutes with analytification in a reasonable sense. – nfdc23 Feb 20 '17 at 5:47