An example of a two-parameter family of affine homogeneous real hypersurfaces in ℂ3

Abstract

This paper studies real hypersurfaces in multidimensional complex spaces homogeneous with respect to holomorphic transformations. In the case of spaces of two complex variables, this problem was completely solved in explicit form as early as in 1932 by Cartan [1]. But even in the problem of describing holomorphic homogeneous real hypersurfaces in complex 3-spaces, there are many questions that remain unanswered so far. Therefore, it is natural to first study the simpler problem of affine homogeneous hypersurfaces in complex space C3. We emphasize that the family of homogeneous real hypersurfaces in C3 is very large (see [2], [3]). For this reason, an important part of the problems stated above is not only finding new families of homogeneous manifolds but also determining their position in some (desirably, complete) classification. Results of the papers [2] and [3] mentioned above show that both the holomorphic and the affine homogeneity of hypersurfaces in multidimensional spaces can be studied within the framework of a unified approach related to local canonical equations of the manifolds under consideration. For example, in [3], by analogy with the well-known (holomorphic)Moser normal form [4], the notion of an affine canonical equation of a real hypersurface in C3 was suggested. According to [3], in a neighborhood of any point, the equation of a strictly pseudoconvex (SPC) real-analytic (not necessarily homogeneous) hypersurface in this space can be reduced by affine transformations to the form