  We prove that any holomorphic locally homogeneous geometric structure on a
complex torus, modelled on a complex homogeneous surface, is translation
invariant. We conjecture that this result is true is any dimension. In higher
dimension we prove it here for nilpotent models. We also prove that in any
dimension the translation invariant $(X, G)$-structures form a union of
connected components in the deformation space of $(X, G)$-structures.
