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

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.