# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Unfolding CR singularities. (English) Zbl 1194.32016

This monograph treats the local equivalence of real submanifolds of complex manifolds under biholomorphic transformations. The real dimension $m$ of the submanifold $M$ under consideration is assumed to be at most equal to the complex dimension $n$ of the ambient complex manifold. The simplest example of this situation are surfaces in ${ℂ}^{2}$ having at some point a complex tangent, which was studied in [J. K. Moser and S. M. Webster, Acta Math. 150, 255–296 (1983; Zbl 0519.32015)]. The points of $M$ to be considered are those carrying CR singularities. The general idea is to find first normal forms, and secondly to investigate how different types of singularities fit into parametrized families of maps (unfoldings). The classification of unfoldings reduces again to normal forms for defining expressions under an appropriate group of transformations.

The starting point for surfaces $M$ in ${ℂ}^{2}$ is the quadric normal form of E. Bishop [Duke Math. J. 32, 1–21 (1965; Zbl 0154.08501)], and some subsequent refinements. Deformation brings the issue of the stability of CR singularities and of properties that are stable under perturbations. One approach uses a Grassmann variety construction to define a general position notion and give an expected codimension for the locus of CR singularities.

To study deformations of $M$ depending on $k$ real parameters, an $\left(m+k\right)$ dimensional real submanifold $\stackrel{^}{M}$ of ${ℂ}^{n+k}$, containing $M$, is introduced. The classification amounts to find normal forms for $\stackrel{^}{M}$ under a group of holomorphic transformations that keeps into account the difference between the coordinates of the original ${ℂ}^{n}$ and the added parameters.

The situation of $n=m$ and $n>m$ are qualitatively different and treated separately. In §6 it is proved (Main Theorem) that, when $M$ is a real analytic submanifold of ${ℂ}^{n}$ of real dimension $m=\frac{2}{3}\left(n+1\right), then there is a coordinate transformation making a non trivial unfolding $\stackrel{^}{M}$ of $M$ real algebraic.

The proofs involve linear approximation and rapid convergence, to solve a system of nonlinear functional equations.

##### MSC:
 32S30 Deformations of singularities (analytic spaces) 58K35 Catastrophe theory 32-02 Research monographs (several complex variables) 32V40 Real submanifolds in complex manifolds
##### Keywords:
CR singularity; normal form; real submanifold