Unfolding CR singularities

Adam Coffman

monograph abstract

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.



temporary Errata information --- I plan to include an official errata
page in some future publication


  Page 9: should be $T_{(\vec0,{\bf0})}\widetilde{M}$

  Page 13: misspelled ``sketched''

  Page 17: Proposition 5.8 has a table with the following two rows:
  
\begin{center}
\rm \small
  \begin{tabular}{|c|c|c|} \hline 
    Label&normal form&comment \\ \hline
     $\begin{array}{c}\mbox{parabolic}\\\!\!\mbox{nondegenerate}\!\!\end{array}$&$\begin{array}{c}h_n=z_1\bar z_1+\frac12(z_1^2+\bar z_1^2)+ic_n(z_1-\bar z_1)x_2\\-i(z_1+\bar z_1)z_1\bar z_1+\eta_n^\sigma x_\sigma z_1\bar z_1+O(4)\\H_\sigma=O(4)\end{array}$&$\begin{array}{c}\gamma=\frac12\\c_n=0,1\\\eta_n^\sigma\mbox{ real}\end{array}$\\ \hline
     $\begin{array}{c}\mbox{parabolic}\\\mbox{degenerate}\end{array}$&$\begin{array}{c}h_n=z_1\bar z_1+\frac12(z_1^2+\bar z_1^2)+ic_n(z_1-\bar z_1)x_2\\+\eta_n^\sigma x_\sigma z_1\bar z_1+O(4)\\H_\sigma=O(4)\end{array}$&$\begin{array}{c}\gamma=\frac12\\c_n=0,1\\\eta_n^\sigma\mbox{ real}\end{array}$\\ \hline
  \end{tabular}
\normalsize
\end{center}

As described in my 2016 AMS talk, the classification of parabolic
normal forms for $n\ge3$ is more accurately summarized in the
following way:

\begin{center}
\rm \small
  \begin{tabular}{|c|c|c|} \hline
    $\begin{array}{c}\mbox{parabolic}\\\!\!\mbox{nondegenerate}\!\!\end{array}$&$\begin{array}{c}h_n=z_1\bar z_1+\frac12(z_1^2+\bar z_1^2)+ic_n(z_1-\bar z_1)x_2\\-i(z_1+\bar z_1)z_1\bar z_1+iK_{\alpha\beta1}\bar z_1x_\alpha
    x_\beta+O(4)\\H_\sigma=O(4)\end{array}$&$\begin{array}{c}\gamma=\frac12\\c_n=0,1\\
    \!K_{\alpha\beta1}\mbox{ real}\!\end{array}$\\ \hline
    $\begin{array}{c}\mbox{parabolic}\\\mbox{degenerate}\end{array}$&$\begin{array}{c}h_n=z_1\bar z_1+\frac12(z_1^2+\bar z_1^2)+ic_n(z_1-\bar z_1)x_2\\+\eta_n^\sigma
    x_\sigma z_1\bar z_1+iK_{\alpha\beta1}\bar z_1x_\alpha
    x_\beta+O(4)\\H_\sigma=O(4)\end{array}$&$\begin{array}{c}\gamma=\frac12\\c_n=0,1\\\eta_n^\sigma\mbox{
    real}\\ \!K_{\alpha\beta1}\mbox{ real}\!\end{array}$\\ \hline
    \end{tabular}
\normalsize
\end{center}

The degenerate/nondegenerate distinction (in the original table and
the corrected version) refers to whether the invariant $\eta$ in the
term $-i\eta(z_1+\bar z_1)z_1\bar z_1$ is $0$ or $1$.  The coefficient $c_n$
is a biholomorphic invariant in either case, while $\eta_n^\sigma$ or
$K_{\alpha\beta1}$ can be further normalized, eliminated, or
re-indexed, depending on the values of the other coefficients and the
dimension $n$.  This update does not affect the table from Proposition
5.1, which referred to $n=2$ only, without any $x$ variables.