Rigid Group Actions on Complex Tori are Projective (after Ekedahl)

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.