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15 July 2016 Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties
Daniel Greb, Stefan Kebekus, Thomas Peternell
Duke Math. J. 165(10): 1965-2004 (15 July 2016). DOI: 10.1215/00127094-3450859

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 Xreg of X to X itself. A simplified version of our main results states that there exists a Galois cover YX, ramified only over the singularities of X, such that the étale fundamental groups of Y and of Yreg agree. In particular, every étale cover of Yreg extends to an étale 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.

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Daniel Greb. Stefan Kebekus. Thomas Peternell. "Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties." Duke Math. J. 165 (10) 1965 - 2004, 15 July 2016. https://doi.org/10.1215/00127094-3450859

Information

Received: 7 July 2014; Revised: 28 July 2015; Published: 15 July 2016
First available in Project Euclid: 5 April 2016

zbMATH: 1360.14094
MathSciNet: MR3522654
Digital Object Identifier: 10.1215/00127094-3450859

Subjects:
Primary: 14J17
Secondary: 14B05 , 14B25 , 14E30

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

Rights: Copyright © 2016 Duke University Press

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Vol.165 • No. 10 • 15 July 2016
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