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.
Best Answer
A reference is Theorem 7.8 of the article by Kollar: "Shafarevich maps and plurigenera of algebraic varieties", Invent. Math. 113. This proves the equality of fundamental groups for quotient singualrities in all dimensions and also for algebraic fundamental groups in the case of klt singularities.