Moduli Spaces of Affine Homogeneous Spaces

Abstract

Apart from global topological problems an affine homogeneous space is locally described by its curvature, its torsion and a slightly less tangible object called its connection in a given base point. Using this description of the local geometry of an affine homogeneous space we construct an algebraic variety $\mathfrak{M}(\mathfrak{gl}V)$, which serves as a coarse moduli space for the local isometry classes of affine homogeneous spaces of dimension dim V. Moreover we associate a $\mathrm{Sym}{V}^{\ast }$-comodule to a point in $\mathfrak{M}(\mathfrak{gl}V)$ and use its Spencer cohomology in order to describes the infinitesimal deformations of this point in the true moduli space ${\mathfrak{M}}_{\infty }(\mathfrak{gl}V)$.