Given a quasiprojective variety with only Kawamata log terminal singularities, we study the obstructions to extending finite étale covers from the smooth locus of to itself. A simplified version of our main results states that there exists a Galois cover , ramified only over the singularities of , such that the étale fundamental groups of and of agree. In particular, every étale cover of extends to an étale cover of . As a first major application, we show that every flat holomorphic bundle defined on extends to a flat bundle that is defined on all of . As a consequence, we generalize a classical result of Yau to the singular case: every variety with at worst terminal singularities and with vanishing first and second Chern class is a finite quotient of an abelian variety. As a further application, we verify a conjecture of Nakayama and Zhang describing the structure of varieties that admit polarized endomorphisms.
"Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties." Duke Math. J. 165 (10) 1965 - 2004, 15 July 2016. https://doi.org/10.1215/00127094-3450859