Let $X$ be a complex manifold with quotient singularities, and let $\tilde X$ be its resolution (that exists, for example, by Hironaka). Then I am pretty sure that $\pi_1(X)\cong \pi_1(\tilde X)$.
- Question. Is there a reference for such a statement? At least in dimension 3?
The reason why this should be true is that it should be possible to find such a resolution of $X$ that the preimage of each point in $\tilde X$ is simply connected ($\mathbb CP^n/G$ is simply-connected for finite $G$). Then everything follows from a standard topological lemma. I guess this statement should be true as well if $X$ is a complex analytic variety with arbitrary Kawamata log terminal singularities (because Fano manifolds are simply connected). I would be grateful for a reference for any kind of such statement.
ADDED. As Francesco says in his answer, this statement is classical for surfaces. I would like to have a reference what would cover at least the case of $3$-folds with non-isolated quotient singularities (say with abelian stabilisers).
UPD. The question is now 100% settled by the reference provided by the answer of Benoit.