A characterization of the finite multiplicity of a CR mapping (Q5961594)

This paper considers the local properties of a CR-mapping, \(f\) between real analytic hypersurfaces, \(M_1, M_2\) in \(\mathbb{C}^n.\) There are two main results: In the first theorem a simple geometric condition, ``\(f^{-1}(f(z_0))\) is finite'' is shown to be equivalent to the complicated algebraic condition, ``\(f:M_1\to M_2\) is of finite multiplicity at \(z_0\)'' provided that 1. \(M_1, M_2\) are real analytic hypersurfaces of essential finite type and 2. \(M_2\) contains no positive dimensional, complex subvarities. In the second theorem it is shown that if \(f^{-1}(f(z_0))\) is finite then \(f\) has a holomorphic extension to a neighborhood of \(z_0\) whenever \(M_1\) is essentially finite and \(M_2\) contains no positive dimensional, complex subvarieties. Various corollaries concerning local properness and existence of holomorphic extensions are also presented. In addition to the elementary properties of holomorphic mappings, the paper relies heavily on results of Bedford and Bell, Baouendi and Rothschild, Trepreau, Tumanov and the ``generalized smoothness principle''.