Analytical Approach to CR Geometry

In the framework of analysis of several complex variables it is natural to identify biholomorphically equivalent geometrical objects. This is appropriate for everything: domains, its boundaries, singular subsets of boundaries (Shilov boundaries), orbits of holomorphic Lie group action, etc.
A germ of a real submanifold in complex space is a highly interesting object. There are three interrelated aspects of this interest: holomorphic automorphisms of the germ, its invariants and classification. These issues belong to CR geometry, which is a domain of interplay between different directions: complex analysis, differential geometry, Lie groups and algebras, theory of differential equations, algebraic geometry, invariant theory, and so on. CR geometry takes its origin in the seminal papers of H. Poincare and E. Cartan. Since then the two approaches in it have been crystallized: analytical, which develops the ideas of Poincare, and geometrical, developing that of Cartan. The author, working in the Poincare paradigm, is going to give a survey of the modern state of the analytical branch of CR geometry.