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Analytic normal form for {CR} singular surfaces in \(\mathbb C^3\). (English) Zbl 1074.32013
The author shows the following:
a) If \(M\) is a real analytic surface in \({\mathbb C}^3\) with a non-degenerate complex tangent at \(p\), then there is a local biholomorphic map \(\varphi\) of \({\mathbb C}^3\), \(\varphi(p)=0\), such that \(\varphi(M)\) is given by \(z_2=\overline{z}_1^2\), \(z_3=z_1\overline{z}_1\).
b) If \(M\) is a real analytic submanifold of \({\mathbb C}^5\) of dimension \(4\) with a non-degenerate complex tangent at \(p\), then there is a local biholomorphic map \(\varphi\) of \({\mathbb C}^5\), \(\varphi(p)=0\), such that \(\varphi(M)\) is given by \(z_5=z_1(\overline{z}_1+x_2+ix_3)\), \(z_4=(\overline{z}_1+x_2+ix_3)^2\), \(y_2=y_3=0\).
The proofs of both theorems involve a rapid iteration argument, by solving a linearized functional equation first. The second normal form is more difficult because of the non-trivial quadratic terms. To avoid the radius of convergence of the linearized functional equation shrinking too much, the author has to find a good solution among all possible ones. This is the main novelty of the paper.
The normal form of real analytic surfaces in \(\mathbb C^2\) has been studied by J. K. Moser and S. M. Webster [Acta Math. 150, 255–296 (1983; Zbl 0519.32015)].

MSC:
32V40 Real submanifolds in complex manifolds
32S05 Local complex singularities
Citations:
Zbl 0519.32015
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