Homogeneous spaces of nonreductive type locally modelling no compact manifold

TL;DR

This paper establishes necessary conditions for compact manifolds modeled on homogeneous spaces, extending previous results to both reductive and nonreductive cases, with applications to coadjoint orbits of solvable algebraic groups.

Contribution

It generalizes earlier conditions for the existence of such manifolds, incorporating nonreductive homogeneous spaces and cohomological methods.

Findings

No compact manifold modeled on certain positive dimensional coadjoint orbits exists.

Provides necessary conditions based on relative Lie algebra cohomology.

Extends previous results to nonreductive homogeneous spaces.

Abstract

We give necessary conditions for the existence of a compact manifold locally modelled on a given homogeneous space, which generalize some earlier results, in terms of relative Lie algebra cohomology. Applications include both reductive and nonreductive cases. For example, we prove that there does not exist a compact manifold locally modelled on a positive dimensional coadjoint orbit of a real linear solvable algebraic group.