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.