On embeddings of homogeneous spaces with small boundary

We study equivariant embeddings with small boundary of a given homogeneous space G/H, where G is a connected linear algebraic group with trivial Picard group and only trivial characters, and H G is an extension of a connected Grosshans subgroup by a torus. Under certain maximality conditions, like completeness, we obtain finiteness of the number of isomorphism classes of such embeddings, and we provide a combinatorial description of the embeddings and their morphisms. The latter allows a systematic treatment of examples and basic statements on the geometry of the equivariant embeddings of a given homogeneous space G/H.