\'Etale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of Abelian varieties

Given a quasi-projective variety X with only Kawamata log terminal singularities, we study the obstructions to extending finite 'etale covers from the smooth locus $X_{\mathrm{r}\mathrm{e}\mathrm{g}}$ of $X$ to $X$ itself. A simplified version of our main results states that there exists a Galois cover $Y \rightarrow X$,ramifiedonlyoverthesingularitiesof$X$, such that the 'etale fundamental groups of $Y$ and of $Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$ agree. In particular, every 'etale cover of $Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$ extends to an 'etale cover of $Y$. As first major application, we show that every flat holomorphic bundle defined on $Y_{\mathrm{r}\mathrm{e}\mathrm{g}}$ extends to a flat bundle that is defined on all of $Y$. As a consequence, we generalise 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 polarised endomorphisms.