Real submanifolds in complex space: polynomial models, automorphisms, and classification problems

Classification of homogeneous CR-manifolds in dimension 4

“…We firstly note that the sphericity of VIa with α = 2, β = 3 follows from [9]. The sphericity of Ic for α = 2 can be verified from the previous fact by applying the binomial formula for (x + iy) 3 . To do the sphericity check for the other surfaces, we refer to the sphericity criterion, formulated in [7].…”

“…As it was explained above, the key point in the obtained classification theorems is the main trichotomy, enabling us to assume that the homogeneity of a non-degenerate 4-dimensional CR-manifold is provided by a 4-dimensional local transitively acting Lie group, which is a priori not clear at all (see, for example, [13]). The main trichotomy, as well as the sphericity criterion for non-degenerate 4-dimensional CR-manifolds in C 3 (see Section 3), are based on the model surface method (see [4,3]). Thus the model surface method for dimension 4 CR-manifolds enables to classify all (sensible) classes of CR-manifolds with symmetries in the dimension under consideration.…”

“…and O (3), O (4) are terms of weights 3 and 4 correspondingly. The manifold, for which the remainders vanish, is called the 4-dimensional CR-cubic (we will call it just the cubic and denote by C ).…”

On holomorphic automorphisms of a class of non-homogeneous rigid hypersurfaces in ℂ N+1

“…It is interesting and important to determine all locally holomorphic automorphisms of a real submanifold in C n (cf. [7]). A criterion for the finite dimensionality of the automorphism group of a hypersurface was given by Stanton [17,18].…”

“…For higher degree model surface, Beloshapka considered the surface Q 3 in the space C n ⊕ C n 2 ⊕ C k with coordinates (z ∈ C n , w 2 ∈ C n 2 , w 3 ∈ C k ), given by the equations Im w 2 = z, z , Im w 3 = 2 Re Φ(z, z), where z, z is an n 2 scalar linearly independent Hermitian form, and Φ(z, z) is a homogeneous C k -valued form of degree three, and gave the structure of the automorphism algebra of the cubic (cf. [7] and references therein). See [6,15] for results for the polynomial models of even higher codimension and degree.…”