Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties

Given a quasiprojective variety X with only Kawamata log terminal singularities, we study the obstructions to extending finite étale covers from the smooth locus Xreg of X to X itself. A simplified version of our main results states that there exists a Galois cover Y→X, ramified only over the singularities of X, such that the étale fundamental groups of Y and of Yreg agree. In particular, every étale cover of Yreg extends to an étale cover of Y. As a first major application, we show that every flat holomorphic bundle defined on Yreg extends to a flat bundle that is defined on all of Y. 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.