What we know about fundamental group of quotient singular manifold? I'm particularly interesting in the case of quotient given by $X:=T/G$ where $T$ is a complex torus and $G$ is finite.
Since $\pi_1$ is a topological objet how the singular of $X$ can affect it? I mean $X$ is endowed with the quotient topology so the loop are defined in the usual way?
But then if I take any resolution $\widetilde{X}$ how $\pi_1(\widetilde{X})$ is related with $\pi_1(X)$? Does this relation depend of the type of singularities (canonical, terminal..)? But I wanna to know it not only in the case of isolated singularities but in the general case.. so maybe it depends only on the codimension of the locus of points with non trivial stabilizer (?)
I know that it can be related with the notion of orbifold, but I don't have too much background on it so if it is necessarily can anyone suggest me a good reference on it?
Thanks!