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Unfolding CR Singularities

About this Title

Adam Coffman, Department of Mathematical Sciences, Indiana University - Purdue University Fort Wayne, 2101 E. Coliseum Blvd., Fort Wayne, Indiana 46805-1499

Publication: Memoirs of the American Mathematical Society
Publication Year: 2010; Volume 205, Number 962
ISBNs: 978-0-8218-4657-5 (print); 978-1-4704-0576-2 (online)
Published electronically: December 1, 2009
MathSciNet review: 2650710
Keywords:CR singularity, normal form, real submanifold
MSC: Primary 32V40; Secondary 32G07, 32S30

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Table of Contents


  • Introduction
  • Topological considerations
  • Local defining equations and transformations
  • A complexification construction
  • Real surfaces in $\mathbb {C}^n$
  • Real $m$-submanifolds in $\mathbb {C}^n$, $m < n$
  • Rapid convergence proof of the main theorem
  • Some other directions


A notion of unfolding, or multi-parameter deformation, of CR singularities of real submanifolds in complex manifolds is proposed, along with a definition of equivalence of unfoldings under the action of a group of analytic transformations. In the case of real surfaces in complex $2$-space, deformations of elliptic, hyperbolic, and parabolic points are analyzed by putting the parameter-dependent real analytic defining equations into normal forms up to some order. For some real analytic unfoldings in higher codimension, the method of rapid convergence is used to establish real algebraic normal forms.

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