Speaker
Description
The problem of describing holomorphically homogeneous real hypersurfaces of spaces $ \mathbb{C}^n$, $n=2,3,4$, is discussed as well as some close questions about the homogeneity of embedded submanifolds. We consider an approach related to the normalization of the surfaces themselves and the Lie algebras of vector fields associated with them, as well as to the determination of these objects through the lower Taylor coefficients of the functions representing them.
The use of canonical (Moser normal) equations of real hypersurfaces in the space $\mathbb{C}^2 $ made it possible to obtain a complete description of holomorphically homogeneous hypersurfaces of this space. By using a similar technique in the space $\mathbb{C}^3 $ a local description was given of fairly representative families of homogeneous Levi-nondegenerate hypersurfaces with ``rich'' symmetry algebras.
Reduction to canonical form of the basic holomorphic vector fields of 5-dimensional Lie algebras in the space $\mathbb{C}^3$ made it possible to obtain a complete description of all holomorphically homogeneous Levi-nondegenerate hypersurfaces with trivial symmetry algebras. A similar prospect emerges for 7-dimensional Lie algebras in the space $\mathbb{C}^4$. At the same time, questions about the dimension of the symmetry algebras of the resulting homogeneous manifolds can also be studied using normal equations, the coefficient approach and computer programs of symbolic calculations.
