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Convergence of formal embeddings between real-analytic hypersurfaces in codimension one. (English) Zbl 1082.32027

Let \(M\subset\mathbb{C}^N\) and \(M'\subset\mathbb{C}^{N'}\) be two strictly pseudoconvex \(C^\omega\) hypersurfaces such that \(N'> N\). A formal embedding mapping \(M\) into \(M'\) and \(p\in M\) into \(p'\in M'\) is a formal holomorphic map \(F: (\mathbb{C}^N, p)\to (\mathbb{C}^{N'}, p')\) such that: i) the pullback by \(F\) of any \(C^\omega\) defining function for \(M'\) in a neighborhood of \(p'\) vanishes on \(M\) as a power series and ii) the differential \(d_pF: T_p(M)\otimes\mathbb{C}\to T_{p'}(M')\otimes\mathbb{C}\) is injective. A conjecture inspired by the classical work of S. S. Chern and J. K. Moser [Acta Math. 133 (1974), 219–271 (1975; Zbl 0302.32015)] in the equidimensional case states that such \(F\) must be convergent, that is, it must be given by the power series of a local holomorphic map. The paper under review gives a positive answer in the case \(N'= N+ 1\).

MSC:

32V40 Real submanifolds in complex manifolds
32V15 CR manifolds as boundaries of domains

Citations:

Zbl 0302.32015
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