## 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

### Keywords:

formal embedding; formal CR transversal map

Zbl 0302.32015
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