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

DOI10.1215/00127094-3450859zbMATH Open1360.14094arXiv1307.5718OpenAlexW1813879881MaRDI QIDQ739072

Author name not available (Why is that?)

Publication date: 17 August 2016

Published in: (Search for Journal in Brave)

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_{m a t h r m r e g}

of

X

to

X

itself. A simplified version of our main results states that there exists a Galois cover

Y i g h t a r r o w X

, ramified only over the singularities of

X

, such that the 'etale fundamental groups of

Y

and of

Y_{m a t h r m r e g}

agree. In particular, every 'etale cover of

Y_{m a t h r m r e g}

extends to an 'etale cover of

Y

. As first major application, we show that every flat holomorphic bundle defined on

Y_{m a t h r m r e g}

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.

Full work available at URL: https://arxiv.org/abs/1307.5718

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