On Orbits of Action of 5-Dimensional Non-Solvable Lie Algebras in Three-Dimensional Complex Space

Abstract

In 1932, E. Cartan described holomorphically homogeneous real hypersurfaces of two-dimensional complex spaces, but a similar study in the three-dimensional case remains incomplete. In a series of works performed by several international teams, the problem is reduced to describing homogeneous surfaces that are nondegenerate in the sense of Levi and have exactly 5-dimensional Lie algebras of holomorphic vector fields. In this paper, precisely such homogeneous surfaces are investigated. At the same time, a significant part of the extensive list of abstract 5-dimensional Lie algebras does not provide new examples of homogeneity. Given in this paper, the complete description of the orbits of 5-dimensional non-solvable Lie algebras in a three-dimensional complex space includes examples of new homogeneous hypersurfaces. These results bring us closer to the completion of a large-scale scientific study that is of interest in various branches of mathematics.