Unfolding CR singularities.

*(English)*Zbl 1194.32016
Mem. Am. Math. Soc. 962, vii, 90 p. (2010).

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 \(\mathbb{C}^2\) 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 \(\mathbb{C}^2\) 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 \((m+k)\) dimensional real submanifold \(\hat{M}\) of \(\mathbb{C}^{n+k}\), containing \(M\), is introduced. The classification amounts to find normal forms for \(\hat{M}\) under a group of holomorphic transformations that keeps into account the difference between the coordinates of the original \(\mathbb{C}^n\) 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 \(\mathbb{C}^n\) of real dimension \(m=\frac{2}{3}(n+1)<n\), then there is a coordinate transformation making a non trivial unfolding \(\hat{M}\) of \(M\) real algebraic.

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

The starting point for surfaces \(M\) in \(\mathbb{C}^2\) 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 \((m+k)\) dimensional real submanifold \(\hat{M}\) of \(\mathbb{C}^{n+k}\), containing \(M\), is introduced. The classification amounts to find normal forms for \(\hat{M}\) under a group of holomorphic transformations that keeps into account the difference between the coordinates of the original \(\mathbb{C}^n\) 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 \(\mathbb{C}^n\) of real dimension \(m=\frac{2}{3}(n+1)<n\), then there is a coordinate transformation making a non trivial unfolding \(\hat{M}\) of \(M\) real algebraic.

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

Reviewer: Mauro Nacinovich (Roma)

##### MSC:

32S30 | Deformations of complex singularities; vanishing cycles |

58K35 | Catastrophe theory |

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

32V40 | Real submanifolds in complex manifolds |

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\textit{A. Coffman}, Unfolding CR singularities. Providence, RI: American Mathematical Society (AMS) (2010; Zbl 1194.32016)

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