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Let $A$ be an Abelian variety (over an algebraically closed field). The group $\mathbb{Z}/2\mathbb{Z}$ acts on $A$ and the quotient is called the Kummer variety of $A$. These are well studied and, as I understand it, their geometry is understood and a classical subject (especially for surfaces).

What happens when you take the quotient of an Abelian variety by other finite groups? Is the geometry of the quotient understood in any sense (e.g., Kodaira dimension)? Are the quotients somehow classified? Is anything else known about them?

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    $\begingroup$ The group of automorphisms of a polarized abelian variety is finite, and Shimura and Taniyama (in their famous 1961 book, p35) define a Kummer variety to be the quotient of a polarized abelian variety by the full group of automorphisms. For a general polarized abelian variety, the automorphism group is Z/2Z, and so I expect that they have been most studied in that case. $\endgroup$
    – JS Milne
    Commented May 14, 2010 at 23:26
  • $\begingroup$ @Prof. Milne: What Shimura and Taniyama do seems natural in the context of CM theory, since it's the higher-dimensional analogue of the Weber function $h: E -> E/\operatorname{Aut}(E,O)$ for a CM elliptic curve. Still, you could ignore the polarization if you want, and then the possibilities for finite subgroups of automorphisms are much more numerous... $\endgroup$
    – Pete L. Clark
    Commented May 15, 2010 at 0:23
  • $\begingroup$ Dear Profs Milne and Clark: Thank you for these responses. And yes, what if I want to ignore polarizations. Are quotients by even just cyclic groups understood? $\endgroup$
    – unknown
    Commented May 17, 2010 at 19:21

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The quotients of abelian surfaces (over $\mathbb C$) by finite groups are classified by Yoshihara. In particular he determines the possible Kodaira dimensions. For instance, if the holonomy part of the group ( quotient by its maximal translation subgroup ) has cardinality greater than $24$ then he shows the quotient is rational.

The precise reference is

  • Yoshihara, Hisao. Quotients of abelian surfaces. Publ. Res. Inst. Math. Sci. 31 (1995), no. 1, 135--143.

Unfortunately, I am not aware of any electronic version of this paper.

For higher dimensional abelian varieties I am not aware of any work studying the finite quotients. But for 3-dimensional complex tori there is for instance this paper by Birkenhake, González-Aguilera, and Lange which classifies the possible finite subgroups. The same authors also have a paper dealing with finite subgroups of the $3$-dimensional abelian varieties (over $\mathbb C$ if I remember correctly).

EDIT (May 18)

You may want to take a look at Complex crystallographic groups I and II. The authors study compact quotients $X$ of $\mathbb C^n$ by discrete subgroups $\Gamma \subset \rm{Aff}(\mathbb C^n)$.

In dimension two they obtain classification results for the pairs $(X, \Gamma)$ assuming $\Gamma$ (more precisely its holonomy part) is generated by reflections (paper I); or $X$ is rational (paper II).

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  • $\begingroup$ This sounds nice - but as you mention I had a difficult time finding the paper! Do you know if their methods can be extended to higher dimensional AVs? $\endgroup$
    – unknown
    Commented May 17, 2010 at 19:22
  • $\begingroup$ I am afraid that a substantial part of the argument reduces to a case-by-case analysis, and the number of cases grows rather quickly with the dimension. $\endgroup$
    – Jorge Vitório Pereira
    Commented May 18, 2010 at 4:51
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There is a classification of finite groups acting freely on abelian threefolds, leading to quotients which are Calabi-Yau threefolds, in Oguiso, Sakurai, Calabi-Yau threefolds of quotient type, arXiv:math/9909175.

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If you are still interested at the moment there is a complete classification in 3 and 4 dimension when the group acts freely on the abelian varieties (they are called hyperelliptic varieties). Some authors are Catanese and Demleitner: if you search their papers you can find others (and better) references.

Maybe you are also interested in the so called generalized kummer.

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    $\begingroup$ Can you give some more precise references? That would be great. $\endgroup$
    – András Bátkai
    Commented May 4, 2021 at 16:59
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    $\begingroup$ In dimension 2 Bagnera-de Franchis give a complete classification of quotient of abelian surface by finite group: you can find this result in Beauville complex Algebraic surfaces List VI.20 they are called bielliptic surface in literature. The seminal works which led to this classification are Bagnera and de Franchis. “Le superficie algebriche le quali ammettono una rappresentazione parametrica mediante funzioni iperellittiche di due argomenti.” and Enqriques and Severi. “Mémoire sur les surfaces hyperelliptiques.” respectively published in 1907 and 1908. $\endgroup$
    – Martina Monti
    Commented May 5, 2021 at 18:05
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    $\begingroup$ In dimension 3 in 1999 Lange in his paper “Hyperelliptic Varieties.” tried to give a classification: his paper is based on the classification of finite automorphism groups acting on 2-dimensional complex tori, given by Fujiki in “Finite Automorphism Groups of Complex Tori of Dimension Two.” , and on the classification of compact groups of affine transformations acting freely and properly discontinuously on $\mathbb{C}^3$, given by Uchida and Yoshihara in "Discontinuous groups of affine transformation of $\mathbb{C}^3$”. $\endgroup$
    – Martina Monti
    Commented May 5, 2021 at 18:10
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    $\begingroup$ I say he tried because he excluded one case: the dihedral group of order 8, but the in 2009 Dekimpe, Halenda and Szczepański, in their article “K$\"a$hler flat manifolds" (see the Table after Theorem 2.4.) proved that the Dihedral group if order 8 defines a free action on abelian threefold so we obtain a smooth quotient. $\endgroup$
    – Martina Monti
    Commented May 5, 2021 at 18:16
  • $\begingroup$ Ops maybe I have to say before.. this classification is about quotient of Abelian Varieties by finite groups which act freely on it and they don't contain translations. They are called Hyperelliptic Varieties. $\endgroup$
    – Martina Monti
    Commented May 5, 2021 at 18:18
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You may be interested in a series of papers I wrote with Giancarlo Lucchini Arteche:

Smooth quotients of abelian varieties by finite groups (A characterization of smooth quotients of abelian varieties of dimension $\geq3$ by finite groups that fix the origin)

Smooth quotients of principally polarized abelian varieties (A characterization of smooth quotients of abelian varieties by finite groups that fix the origin as well as a principal polarization; this gives a moduli-theoretic spin on the problem)

Smooth quotients of complex tori by finite groups (with an appendix by Stephen Griffeth) (Here we provide a "bridge" between the case of smooth quotients where the group fixes the origin and the case of a free action. Essentially general smooth quotients of complex tori by finite groups are fibrations in products of projective spaces over an étale quotient of a complex torus)

There is a lot to be done in the case of étale quotients, and as the other posts say, much work has been done in this case by Uchida and Yoshihara, Lange, Catanese and Demleitner.

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