Fundamental group of a compact branched cover

Ciao Francesco.

The more general version of this Theorem that I know is in

Fox, Ralph H. Covering spaces with singularities 1957 A symposium in honor of S. Lefschetz pp. 243–257 Princeton University Press, Princeton, N.J.

see the Theorem at page 254, for branched covering of topological spaces.

The proof follows the same lines of the answer by Moishe Kohan.

Consider a small complex 1-dimensional disk $Dsubset Y$ transversal to $B$ and let $c$ denote the image in $pi=pi_1(Y-B)$ of the oriented loop $partial D$. Let $n$ denote the order of the image of $c$ under $varphi$. Now, form a complex orbifold ${mathcal O}$ (a stack in your language) with the underlying space $Y$ and orbi-data ${mathbb Z}/n$ along $B$. (I am sure, you know what I mean.) Then $$ pi_1({mathcal O})cong pi/ langle c^nrangle^{pi}, $$ where $langle c^n rangle^{pi}$ denotes the normal closure of the subgroup $langle c^n rangle$ in $pi$. The homomorphism $varphi$ descends to a homomorphsm
$$ psi: pi_1({mathcal O})to G. $$ Then $pi_1(X)$ is isomorphic to the kernel of $psi$.

In fact, this works in much greater generality, as the divisor $B$ need not be a smooth submanifold and need not be connected, but instead of a single disk $D$ you have to take a collection of disks transversal to the components of the smooth locus of $B$.