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Analytic normal form for {CR} singular surfaces in $\Bbb C^3$. (English) Zbl 1074.32013
The author shows the following: a) If $M$ is a real analytic surface in ${\Bbb C}^3$ with a non-degenerate complex tangent at $p$, then there is a local biholomorphic map $\varphi$ of ${\Bbb C}^3$, $\varphi(p)=0$, 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 ${\Bbb C}^5$ of dimension $4$ with a non-degenerate complex tangent at $p$, then there is a local biholomorphic map $\varphi$ of ${\Bbb C}^5$, $\varphi(p)=0$, 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 $\Bbb C^2$ has been studied by {\it J. K. Moser} and {\it S. M. Webster} [Acta Math. 150, 255--296 (1983; Zbl 0519.32015)].

##### MSC:
 32V40 Real submanifolds in complex manifolds 32S05 Local singularities (analytic spaces)