On the Problem of Describing Holomorphically Homogeneous Real Hypersurfaces of Four-Dimensional Complex Spaces

We discuss two fragments of a large problem that extends the author's recently completed similar studies in the space $\mathbb C^3$ to the next dimension. One of the fragments is related to the local description of nonspherical holomorphically homogeneous strictly pseudoconvex hypersurfaces in $\mathbb C^4$ with stabilizers of submaximal dimension. Using the Moser normal form technique and the properties of subgroups of the unitary group $\mathrm U(3)$, we show that up to holomorphic equivalence there exist only two such surfaces. Both of them are natural generalizations of known homogeneous hypersurfaces in the space $\mathbb C^3$. In the second part of the paper, we consider a technique of holomorphic realization in $\mathbb C^4$ of abstract seven-dimensional Lie algebras that correspond, in particular, to homogeneous hypersurfaces with trivial stabilizer. Some sufficient conditions for the Lie algebras are obtained under which the orbits of all realizations of such algebras are Levi degenerate. The schemes of studying holomorphically homogeneous hypersurfaces that were used in the two-dimensional (É. Cartan) and three-dimensional (Doubrov, Medvedev, and The; Fels and Kaup; Beloshapka and Kossovskiy; Loboda) situations and resulted in full descriptions of such hyperdurfaces turn out to be quite efficient in the case of greater dimension of the ambient space as well.