Comparing fundamental groups of a complex orbifolds and their resolutions.

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.

The proof of simple connectedness of a resolution of quotient singularities is in this paper of mine http://arxiv.org/abs/math/9903175 Theorem 4.1 (published in Asian J. Math. 4, 2000, no. 3, 553-563), but I guess it was known to Bogomolov long ago (and maybe even published in some of his papers)

Dear Dmitri, the result you hope is also true (ie when X has only klt singularities, its fundamental group is isomorphic to the one of any of its desingularization). This theorem is due to Takayama (Local simple connectedness of resolution of log-terminal singularities, International journal of Math 2003).

Bests, Benoît