Towards a Classification of Rigid Product Quotient Varieties of Kodaira Dimension 0

TL;DR

This paper classifies rigid quotients of products of elliptic curves by finite groups, identifying conditions for free actions and providing a complete classification of certain dimensions and groups.

Contribution

It offers a complete classification of singular and smooth quotients of elliptic curve products under rigid group actions, highlighting specific groups and dimensions where free actions occur.

Findings

Only for G=He(3) or Z_3^2 and dimension ≥4 can the action be free.

Complete classification of singular quotients in dimension 3.

Complete classification of smooth quotients in dimension 4.

Abstract

In this paper the authors study quotients of the product of elliptic curves by a rigid diagonal action of a finite group $G$. It is shown that only for $G = \operatorname{He(3)}, \mathbb Z_3^2$, and only for dimension $\geq 4$ such an action can be free. A complete classification of the singular quotients in dimension 3 and the smooth quotients in dimension $4$ is given. For the other finite groups a strong structure theorem for rigid quotients is proven.