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

TL;DR

This paper proves that complex tori with rigid group actions are necessarily projective and describes their structure via CM-fields, also showing all such actions can deform to projective ones.

Contribution

It provides a detailed proof that rigid group actions on complex tori imply projectivity and characterizes their structure using CM-fields, expanding understanding of complex tori symmetry.

Findings

Rigid group actions on complex tori imply projectivity.

Complex tori with such actions are structured via CM-fields.

All group actions on complex tori can deform to projective actions.

Abstract

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.