Symmetries of holomorphic geometric structures on tori

TL;DR

This paper proves that holomorphic locally homogeneous geometric structures on complex tori are translation invariant, and extends some results to higher dimensions and nilpotent models, analyzing deformation spaces.

Contribution

It establishes translation invariance for such structures on tori and explores their deformation space structure, including higher-dimensional cases and nilpotent models.

Findings

Holomorphic locally homogeneous structures on tori are translation invariant.

Translation invariant structures form connected components in deformation spaces.

Results extended to higher dimensions for nilpotent models.

Abstract

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.