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

Abstract

Given a quasi-projective 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 $Y\to X$, 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 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.

v1: this article supersedes arXiv:1302.1655; v2: slightly more general results, simplified topological arguments in Part II of the paper, improved exposition; v3: improved exposition, minor changes as requested by referee; to appear in Duke Mathematical Journal

References in corpus (2)

• Revêtements étales et groupe fondamental (SGA 1)
• Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of Abelian varieties

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