Moduli Spaces of Affine Homogeneous Spaces

Abstract

Every affine homogeneous space is locally described by its curvature, its torsion and a slightly less tangible object, its connection. Using this description of the local geometry of an affine homogeneous space we construct a variety $\mathfrak{M}(\,\mathfrak{g}\,V\,)$, which serves as a coarse moduli space for the local isometry classes of affine homogeneous spaces. Infinitesimal deformations of an isometry class of affine homogeneous spaces in this moduli space are \linebreak described by the Spencer cohomology of a comodule associated to a point in $\mathfrak{M}_\infty(\,\mathfrak{g}\,V\,)$. In an appendix we discuss the relevance of this construction to the study of locally homogeneous spaces.