Erratum to: A note on Galois embeddings of abelian varieties

Auffarth, Robert

doi:10.1007/s00229-018-1080-6

Cited by 1 publication

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“…It seems natural to think that G/T G ∼ = G 0 acts on A/T G by automorphisms that fix the origin. Unfortunately this is not true, and has been mistakenly used in [Yos07] and [Auf18] (for the latter article a subsequent errata [Auf19] was published that did not rely on this problem). Actually, the actions of G/T G and G 0 can be quite Key words and phrases.…”

Section: Introductionmentioning

confidence: 99%

Smooth quotients of abelian varieties by finite groups

Auffarth¹,

Arteche²

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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Let A be an abelian variety and G a finite group acting on A without translations such that A/G is smooth. Consider the subgroup F ≤ G generated by elements fixing at least a point. We prove that there exists a point x ∈ A fixed by the whole group F and that the quotient A/G is a fibration of products of projective spaces over anétale quotient of an abelian variety (theétale quotient being Galois with group G/F ). In particular, when G = F , we may assume that G fixes the origin, and this reduces the classification of smooth quotients to previous work by the authors, where the case of actions fixing the origin was treated.An ingredient of the proof of our fixed-point theorem is a result proving that in every irreducible complex reflection group there is an element which is not contained in any proper reflection subgroup and that Coxeter elements have this property for well-generated groups. This result is proved by Stephen Griffeth in an appendix. MSC codes: primary 14L30, 14K99; secondary 20F55.

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Section: Introductionmentioning

confidence: 99%