For smooth algebraic varieties $X$ over $\mathbb{C}$, the Riemann existence theorem establishes an equivalence of categories between the category of finite etale covers of $X$ and finite unramified covers of $X^\text{an}$.

If $X$ is instead a Deligne Mumford stack over Spec $\mathbb{C}$ corresponding to a complex orbifold $X^\text{an}$, is there an analogous theorem providing an equivalence between finite etale morphisms from DM stacks to $X$ and finite orbifold coverings of $X^\text{an}$? (hence etale fundamental group of such a DM stack is just the profinite completion of the orbifold fundamental group?)

In particular, I'm thinking of the situation where the stack is the moduli stack of elliptic curves over $\mathbb{C}$ and the orbifold is just $\mathcal{H}/SL_2(\mathbb{Z})$.

Can anyone provide a reference to a precise statement?

  • $\begingroup$ A finite etale cover of a DM stack is "the same" as descent datum on a finite etale cover of a scheme that is etale over the DM stack, so what is the difficulty in pulling down the result from the scheme case via descent theory? $\endgroup$ – user74230 Dec 20 '14 at 6:01

The answer is yes. See Theorem 20.4 on p. 80 in Behrang Noohi's



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.