On holomorphic homogeneity of real hypersurfaces of general position in \(\mathbb{C}^3\)
The article deals with the equivalence problem for homogeneous real-analytic hypersurfaces in \(\mathbb C^3\). A solution has been provided in the Levi degenerate case by \textit{G. Fels} and \textit{W. Kaup} [Acta Math. 201, No. 1, 1--82 (2008; Zbl 1171.32023)]. In the strictly pseudoconvex case, which is the setting of the reviewed paper, the second author in [Zbl 1023.32025; Zbl 1023.32025] gave results when there is a large isotropy group and another instance in [Zbl 1060.32021)]. For coordinates \((z_1,z_2,w)\) of \(\mathbb C^3\) write \(z=(z_1,z_2) \in \mathbb C^2\) and \(w=u+iv\). According to the result of \textit{S. S. Chern} and \textit{J. K. Moser} [Usp. Mat. Nauk 38, No. 2(230), 149--193 (1983; Zbl 0519.32014)] any strictly pseudoconvex real-analytic hypersurface in \(\mathbb C^3\) is biholomorphically equivalent to a hypersurface given by \[ \begin{aligned} v = |z_1|^2 + |z_2|^2 + \sum_{k,\ell \geq 2,m\geq 0} N_{k\ell m}(z,\bar z) u^m, \end{aligned} \] where \(N_{k\ell m}\) is a homogeneous polynomial of degree \(k\) in \(z\) and degree \(\ell\) in \(\bar z\) and the polynomials \(N_{22m}, N_{32m}\) and \(N_{33m}\) satisfy certain trace conditions. It was shown in [\textit{V. V. Ezhov} et al., Math. Notes 66, 3--4, 513--515 (1999); translation from Mat. Zametki 66, 4, 624--626 (1999; Zbl 1395.32020)] that after a change of coordinates preserving the Chern-Moser normal form, \(N_{220}\) can be assumed to be one of two specific polynomials. In the paper under review one case is considered and the corresponding hypersurface is referred to as being of ``general position''. The reader is referred to the paper for a detailed description. It was shown by the second author [Sb. Math. 192, No. 12, 1741--1761 (2001; Zbl 1029.32021); translation from Mat. Sb. 192, No. 12, 3--24 (2001)] that in the case of a homogeneous hypersurface, the above given Chern-Moser normal form is uniquely determined by the polynomials \(N_{220},N_{221},N_{222},N_{320},N_{321},N_{330}\) and \(N_{420}\). This set of polynomials depends on at most 100 real coefficients. The present work analyzes the linear equations obtained from the tangency of vector fields, whose existence is guaranteed by homogeneity. This allows to gain new equations and to reduce the number of coefficients (partly with the aid of a computer algebra system). The main result is that the dimension of the moduli space for the class of homogeneous strictly pseudoconvex real-analytic hypersurfaces of general position in \(\mathbb C^3\) is bounded by \(13\). In the last section for the class of tubes over affine homogeneous surfaces in \(\mathbb R^3\) the authors provide some of the higher-order polynomials appearing in the normal form and discuss their approach in this particular case.
scientific article; zbMATH DE number 6844172
scientific article