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

TL;DR

This paper investigates the extension of étale covers over Kawamata log terminal spaces, showing that certain covers can be extended across singularities, leading to new insights on flat bundles and the structure of varieties with special properties.

Contribution

It proves the existence of specific Galois covers that align étale fundamental groups of the cover and its smooth locus, and applies this to extend flat bundles and classify certain singular varieties.

Findings

Existence of Galois covers matching étale fundamental groups of the cover and smooth locus

Extension of flat holomorphic bundles from smooth to entire space

Characterization of varieties with terminal singularities as quotients of Abelian varieties

Abstract

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 $X_{\mathrm{reg}}$ of $X$ to $X$ itself. A simplified version of our main results states that there exists a Galois cover $Y \rightarrow X$, ramified only over the singularities of $X$, such that the \'etale fundamental groups of $Y$ and of $Y_{\mathrm{reg}}$ agree. In particular, every \'etale cover of $Y_{\mathrm{reg}}$ extends to an \'etale cover of $Y$. As first major application, we show that every flat holomorphic bundle defined on $Y_{\mathrm{reg}}$ 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.