Smooth quotients of abelian varieties by finite groups

Abstract

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.

Q-abelian and $\mathbb Q$-Fano finite quotients of abelian varieties

“…Yoshihara [Yos95] gave a classification of finite quotients of abelian surfaces. Auffarth-Lucchini Arteche [ALA20] and Auffarth-Lucchini Arteche-Quezada [ALAQ18] proved that, if a finite group action on an abelian variety fixes the origin and induces an irreducible action on the tangent space at the origin, then the quotient is a projective space. Kollár-Larsen [KL09] proved that any finite quotient of a simple abelian variety of dimension ≥ 4 is non-uniruled.…”

Fibrations associated to smooth quotients of abelian varieties

“…In [ALA20a] R. Auffarth and G. Lucchini Arteche study smooth quotients of abelian varieties by finite groups whose action fixes the origin. Let A be an abelian variety and G a finite group of automorphisms of A fixing the origin such that A/G is smooth.…”

“…The action of G on A induces an action of G on P G . There exists a fibration 1 A/G → A/P G and the fibers are isomorphic to P G /G and smooth (Proposition 2.9 [ALA20a]).…”

“…If A 0 = 0, or equivalently A = P G , the fibration A/G → A/P G ∼ = 0 has just one fiber and this corresponds to A/G. This case was studied in [ALA20a] for dimension ≥ 3, and the case of dimension 2 was studied in [ALAQ19] and can be summarized in the following two results: Theorem 1.1 (Theorem 1.1 [ALA20a] -Theorem 1.1 [ALAQ19]). Let A be an abelian variety and let G be a (non trivial) finite group of automorphisms of A that fix the origin.…”

“…Even if A 0 = 0, the analytic representation of G on A might not always be irreducible. However, in this case the abelian variety A and the group G can be conveniently decomposed as in the next result: Theorem 1.2 (Theorem 1.3 [ALA20a]). Let A be an abelian variety and let G be a finite subgroup of Aut 0 (A) such that A/G is smooth.…”