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Analytic normal form for CR singular surfaces in 3 . (English) Zbl 1074.32013

The author shows the following:

a) If M is a real analytic surface in 3 with a non-degenerate complex tangent at p, then there is a local biholomorphic map φ of 3 , φ(p)=0, such that φ(M) is given by z 2 =z ¯ 1 2 , z 3 =z 1 z ¯ 1 .

b) If M is a real analytic submanifold of 5 of dimension 4 with a non-degenerate complex tangent at p, then there is a local biholomorphic map φ of 5 , φ(p)=0, such that φ(M) is given by z 5 =z 1 (z ¯ 1 +x 2 +ix 3 ), z 4 =(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 2 has been studied by J. K. Moser and S. M. Webster [Acta Math. 150, 255–296 (1983; Zbl 0519.32015)].

MSC:
32V40Real submanifolds in complex manifolds
32S05Local singularities (analytic spaces)