  In this note we give a detailed proof of a result (sketched by Torsten
Ekedahl in a discussion with the first author several years ago), describing
complex tori admitting a rigid group action and showing explicitly their
projectivity and their structure in terms of CM-fields. In the appendix, joint
with Benoit Claudon, we show, using a method of Green-Voisin, that all group
actions on complex tori deform to projective ones.
