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?
Best Answer
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
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).