Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of Abelian varieties
By D. Greb, Stefan Kebekus, T. Peternell. 2013
152 citations recorded by the source corpus.
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 $X_{\mathrm{reg}}$ 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 \'etale fundamental groups of $Y$ and of $Y_{\mathrm{reg}}$ agree. In particular, every \'etale cover of $Y_{\mathrm{reg}}$ extends to an \'etale 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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/bd7364226f716eb24b0f80197deb43e90058cec9
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- The local fundamental group of a Kawamata log terminal singularity is finite (2020, background): “In contrary to the finite versions from [34], we have no idea if these statements could be of any use.”
- The dual complex of Calabi–Yau pairs (2015, unknown): “The finiteness of π̂1 ( X sm ) follows from [11,36]; it is conjectured thatπ1 ( X sm )”