Duke Mathematical Journal

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

Daniel Greb, Stefan Kebekus, and Thomas Peternell

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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

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

Received: 7 July 2014
Revised: 28 July 2015
First available in Project Euclid: 5 April 2016

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Mathematical Reviews number (MathSciNet)

Primary: 14J17: Singularities [See also 14B05, 14E15]
Secondary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 14E30: Minimal model program (Mori theory, extremal rays) 14B25: Local structure of morphisms: étale, flat, etc. [See also 13B40]

minimal model program algebraic fundamental group KLT singularities flat vector bundles torus quotients polarized endomorphisms


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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