Monge-Amp\`ere exhaustions of almost homogeneous manifolds

We consider three fundamental classes of compact almost homogeneous manifolds and show that the complements of singular complex orbits in such manifolds are endowed with plurisubharmonic exhaustions satisfying complex homogeneous Monge-Amp`ere equations. This extends to a new family of mixed type examples various classical results on parabolic spaces and complexifications of symmetric spaces. Rigidity results on complex spaces modeled on such new examples are given.