Homogeneous real hypersurfaces in $\mathbb C^3$: completion of classification

The complete classification of holomorphically homogeneous real hypersurfaces in two-dimensional complex spaces (in local and global forms) was proposed by E. Cartan in 1932.
Final classification of locally homogeneous hypersurfaces in complex spaces of the following dimension essentially uses the properties of the degeneracy or non-degeneracy of Levi form of the surfaces under study. An important role is also played by the dimensions estimate $ 5 \le \dim g \le 8 $ (Loboda-2000) for Lie algebras of holomorphic vector fields on non-degenerate Levi non-spherical homogeneous surfaces.
In accordance with this estimate, all homogeneous nondegenerate surfaces with 8- and 7-dimensional (Loboda 2001), and then with 6-dimensional Lie algebras (Dubrov-Medvedev-The-2017) were described. Levi-degenerate homogeneous surfaces were completely studied in 2008 by Fels and Kaup.
The report discusses the final part of classification associated with non-degenerate homogeneous surfaces having only trivial stabilizers. Individual blocks of the volumetric classification of abstract 5-dimensional Lie algebras (Mubarakzyanov-1961) were associated with the problem of homogeneity by studying holomorphic realizations of such algebras.
In addition to the previously known homogeneous manifolds, there are only three (cited in the report) new types of holomorphically homogeneous real hypersurfaces. All these new examples are indefinite (non-degenerate) surfaces. In general, the studied family of homogeneous hypersurfaces splits into 40 types of manifolds; many of them, but not all, are holomorphically equivalent to tubular manifolds. The maximum dimension of the moduli spaces for the individual components of this family is 2.
The presented results were obtained jointly with Akopyan R.S., Atanov A.V., Kossovskiy I.G.
The study was supported by the RFBR grant № 17-01-00592-a.