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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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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. – JS Milne May 14 '10 at 23:26
@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... – Pete L. Clark May 15 '10 at 0:23
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? – unknown May 17 '10 at 19:21

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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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? – unknown May 17 '10 at 19:22
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. – Jorge Vitório Pereira May 18 '10 at 4:51

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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