Smooth quotients of complex tori by finite groups (with an appendix by Stephen Griffeth)

TL;DR

This paper studies smooth quotients of complex tori by finite groups, proving the existence of fixed points and describing the structure of the quotient as a fibration, thereby reducing classification problems to simpler cases.

Contribution

It generalizes previous results on abelian varieties to complex tori and introduces a fixed-point theorem crucial for understanding smooth quotients.

Findings

Existence of a fixed point for the subgroup generated by elements with fixed points.

Structure of the quotient as a fibration over an étale quotient of a complex torus.

Reduction of classification problems to étale quotients.

Abstract

Let $A$ be a complex torus and $G$ a finite group acting on $A$ without translations such that $A/G$ is smooth. Consider the subgroup $F\leq G$ generated by elements that have at least one fixed point. We prove that there exists a point $x\in A$ fixed by the whole group $F$ and that the quotient $A/G$ is a fibration of products of projective spaces over an \'etale quotient of a complex torus (the \'etale quotient being Galois with group $G/F$). In particular, when $G=F$, we may assume that $G$ fixes the origin. This is related to previous work by the authors, where the case of actions on abelian varieties fixing the origin was treated. Here, we generalize these results to complex tori and use them to reduce the problem of classifying smooth quotients of complex tori to the case of \'etale quotients. 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.