  Let A be a self-injective algebra over an algebraically closed field k. We
show that if an A-module M of complexity one has an open orbit in the variety
of d-dimensional A-modules, then M is periodic. As a corollary we see that any
simple A-module of complexity one must be periodic. In the course of the proof,
we also show that modules with open orbits are preserved by stable equivalences
of Morita type between self-injective algebras.
