Homogeneous Surfaces in $$\mathbb C^4$$ Associated with a 5-Dimensional Completely Nondegenerate Cubic Model Surface of
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Homogeneous Surfaces in C4 Associated with a 5-Dimensional Completely Nondegenerate Cubic Model Surface of CR-Type (1, 3) V. K. Beloshapka∗,1 and M. Sabzevari∗∗,2 ∗
Department of Mechanics and Mathematics, Moscow State University, Vorob’evy gory, 119991, Moscow, Russia ∗∗ Department of Mathematics, Shahrekord University, 88186-34141, Shahrekord, Iran and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), 19395-5746, Tehran, Iran E-mail: 1 [email protected], 2 [email protected] Received February 26, 2020; Revised February 27, 2020; Accepted March 30, 2020
Abstract. The action on the space C4 of the 7-dimensional Lie group of infinitesimal holomorphic automorphisms of a completely nondegenerate cubic model surface Q of CR-type (1, 3) is considered. All orbits of the given action are found and their biholomorphic classification is given. One of the orbits coincides with the surface Q (the 5-dimensional orbit), two orbits are 6-dimensional, and the remaining part of the space C4 , the complement to the above orbits, foliates into 7-dimensional real orbits. It is also proved that the algebra of infinitesimal holomorphic automorphisms of all orbits, except for the twelve holomorphically degenerate orbits, coincides with the algebra of the surface Q. DOI 10.1134/S1061920820040020
Let Mξ be the germ of a smooth real generating subvariety of the space CN . Let n be its CR-dimension and let d be the codimension. In this case, n + d = N . We call the pair (n, d) the CR-type. Let aut Mξ be the Lie algebra consisting of germs of real vector fields generating one-parameter groups of holomorphic transformations in a neighborhood of the point ξ that preserve Mξ ; let autξ Mξ be the subalgebra of aut Mξ consisting of the fields X ∈ aut Mξ such that X(ξ) = 0; let Aut Mξ be the local group generated by the fields in aut Mξ ; let Autξ Mξ consist of transformations φ ∈ Aut Mξ such that φ(ξ) = ξ. If Mξ is of finite Bloom–Graham type, then one can associate with this germ a real algebraic surface Q of the same type, which is the tangent model surface [5]. The model surfaces are interesting for many reasons. For example dim Aut Mξ dim Aut Q. In [3, 4], the action of the group of holomorphic automorphisms of a model surface of CR-type (1, 2) in C3 was studied and all its orbits were described. In this paper, we consider a similar question related to the space C4 . Namely, for the model surface Q of CR-type (1, 3), we study the action of the 7-dimensional group of its automorphisms Aut Q on the space C4 and calculate all its orbits. Up to biholomorphic equivalence, there is only one completely nondegenerate 5-dimensional model surface Q of type (1, 3). In the coordinates (z, wj = uj + ivj ), j = 1, 2, 3, of the space C4 , Q is given by the relations ⎧ ⎨ v1 = zz, v2 = z 2 z + zz 2 , (1) ⎩ v3 = −i (z 2 z − zz2 ). The local group of all holomorphic transformations of C4 that are invertible at the origin and preserve the germ Q at the origin is the 7-dimensional Lie group Aut Q of triangu
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