ENTALE FUNDAMENTAL GROUPS OF KAWAMATA LOG TERMINAL SPACES, FLAT SHEAVES, AND QUOTIENTS OF ABELIAN VARIETIES

摘要

Given a quasiprojective variety X with only Kawamata log terminal singularities, we study the obstructions to extending finite etale covers from the smooth locus X-reg 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 etale fundamental groups of Y and of Y-reg agree. In particular, every etale cover of Y-reg extends to an etale 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.