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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
Zbl 0519.32015