## Duke Mathematical Journal

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

#### Abstract

Given a quasiprojective variety $X$ with only Kawamata log terminal singularities, we study the obstructions to extending finite étale 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 étale fundamental groups of $Y$ and of $Y_{\mathrm{reg}}$ agree. In particular, every étale cover of $Y_{\mathrm{reg}}$ extends to an étale cover of $Y$. As a 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 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.

#### Article information

Source
Duke Math. J., Volume 165, Number 10 (2016), 1965-2004.

Dates
Revised: 28 July 2015
First available in Project Euclid: 5 April 2016

https://projecteuclid.org/euclid.dmj/1459878278

Digital Object Identifier
doi:10.1215/00127094-3450859

Mathematical Reviews number (MathSciNet)
MR3522654

Zentralblatt MATH identifier
1360.14094

#### Citation

Greb, Daniel; Kebekus, Stefan; Peternell, Thomas. Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties. Duke Math. J. 165 (2016), no. 10, 1965--2004. doi:10.1215/00127094-3450859. https://projecteuclid.org/euclid.dmj/1459878278

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