  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.
