arXiv subjects
Publications and source records attributed to V. K. Beloshapka.
The orbits of exceptional CR-quadrics are studied. It is proved that the orbit of the exceptional (4,4)-quadric constructed in [1] is isolated. Using the algebraization procedure described in [2], it is shown that the space of orbits of exceptional quadrics of type (28,28) has positive dimension.
Exceptional CR-quadrics are studied. An example of an exceptional (4,4)-quadric is constructed; it realizes the minimum exceptional type with respect to both n and k. Its graded Lie algebra is described. The available information on exceptional types is summarized, and the lattice of CR-types is decomposed into the disjoint union of three sets: A, the types for which exceptional quadrics are impossible; B, the types for which examples of exceptional quadrics are known; and C, the types whose status is currently unknown (neither an example nor a nonexistence proof is known). Several questions are posed.
In the paper, a systematic construction of the theory of "weighted" model surfaces is given. This construction is based on the notion of the Bloom-Graham-Stepanova type. The key instrument is the Poincare construction. It is shown how weighted approach extends possibilities of the model surface method. New questions are formulated.
It is proved that the graded Lie algebras of infinitesimal holomorphic automorphisms of nondegenerate quadric of codimension k do not have nonzero graded components of weight greater than 2k. It is also proved that for $k = 3$ and for RAQ-quadrics there are no components of weight greater than 2.
A generalization of the homological Poincare's operator was used to estimate the dimension of the Lie algebra of infinitesimal holomorphic automorphisms of an arbitrary germ of a real analytic hypersurface in ${\bf C}^4$. The following alternative is proved: either this dimension is infinite, or it does not exceed 24. Value 24 takes place only for one of two nondegenerate hyperquadrics. If the hypersurface is 2-nondegenerate at a generic point, then the dimension does not exceed 17, and if the hypersurface is 3-nondegenerate at a generic point, then the estimate is 20.
It is proved that the group of holomorphic automorphisms of holomorphically homogeneous nondegenerate (finite Bloom-Graham type + holomorphic nondegenaracy) model surface Q is a subgroup of the group of birational automorphisms of the ambient space (Cremona group) with uniformly bounded degree. An estimate of the degree of the automorphisms in terms of the dimension of the ambient space is given (Theorem 4). It is proved that none of the conditions of the theorem can be omitted. We consider also the question about connectedness of the automorphism group of Q (Theorem 7).
Locally homogeneous CR-manifolds in dimension 3 were classified, up to local CR-equivalence, by E.Cartan. We classify, up to local CR-equivalence, all locally homogeneous CR-manifolds in dimension 4. The classification theorem enables us also to classify all symmetric CR-manifolds in dimension 4, up to local biholomorphic equivalence. We also prove that any 4-dimensional real Lie algebra can be realized as an algebra of affine vector fields in a domain in $\CC{3}$, linearly independent at each point.
The model 4-dimensional CR-cubic in $\CC{3}$ has the following "model" property: it is (essentially) the unique locally homogeneous 4-dimensional CR-manifold in $\CC{3}$ with finite-dimensional infinitesimal automorphism algebra $\mathfrak{g}$ and non-trivial isotropy subalgebra. We study and classify, up to local biholomorphic equivalence, all $\mathfrak{g}$-homogeneous hypersurfaces in $\CC{3}$ and also classify the corresponding local transitive actions of the model algebra $\mathfrak{g}$ on hypersurfaces in $\CC{3}$.