On Degenerate Orbits of Real Lie Algebras in Multidimensional Complex Spaces

Abstract

The article studies (locally) holomorphically homogeneous real hypersurfaces of complex spaces. Currently, the problem of classifying such hypersurfaces is completely solved only in the spaces \(\mathbb{C}^{2}\) and \(\mathbb{C}^{3}\). As the dimension of the ambient space grows, so does the relative part of Levi-degenerate manifolds in the family of all homogeneous hypersurfaces. In particular, this family includes holomorphically degenerate hypersurfaces, which are (locally) direct products of homogeneous hypersurfaces from spaces of smaller dimensions and spaces \(\mathbb{C}^{k}\). The article proves a sufficient Levi-degeneracy condition of all orbits in spaces \(\mathbb{C}^{n+1}\) \((n \ge 3)\) for \((2n+1)\)-dimensional Lie algebras of holomorphic vector fields having full rank at the points in \(\mathbb{C}^{n+1}\). The proven condition is the existence of an abelian subalgebra of codimension 2 in the Lie algebra under discussion. It is shown that in the case \(n = 3\), this condition holds for a large family of 7-dimensional Lie algebras. Holomorphically homogeneous hypersurfaces, i.e. the orbits of these algebras in \(\mathbb{C}^{4}\) can only be Levi-degenerate manifolds. We provide an example of a family of 7-dimensional Lie algebras that have 5-dimensional abelian ideals and Levi-degenerate (but not holomorphically degenerate) orbits.

DOI 10.1134/S1061920823040027