Moduli Spaces of Affine Homogeneous Spaces

TL;DR

This paper constructs an algebraic variety serving as a coarse moduli space for local isometry classes of affine homogeneous spaces, using curvature, torsion, and connection data, and studies their infinitesimal deformations via Spencer cohomology.

Contribution

It introduces a new algebraic variety as a coarse moduli space for affine homogeneous spaces and links their local geometric data to algebraic and cohomological structures.

Findings

Construction of the algebraic variety (\u1e9f V) as a moduli space.

Association of a V^*-comodule to points in the moduli space.

Use of Spencer cohomology to describe infinitesimal deformations.

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 $\mathrm{Sym}V^*$-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 $\mathfrak{M}_\infty(\mathfrak{gl}\,V\,)$.