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Convergence and finite determination of formal CR mappings. (English) Zbl 0958.32033
The authors first give basic definitions, for formal holomorphic mappings, such as manifold ideal, formal real submanifold of \(\mathbb{C}^N\) through 0, formal mapping \(F:(\mathbb{C}^k,0) \to(\mathbb{C}^m,0)\), generic submanifolds of \(\mathbb{C}^N\) as well as a formal mapping which maps \(M\) into \(M'\), where \(M\) and \(M'\) are formal real submanifolds through 0 in \(\mathbb{C}^N\) and \(\mathbb{C}^{N'}\) respectively. The authors study the convergence and finite determination of formal holomorphic mappings of \((\mathbb{C}^N,0)\) mapping one real submanifold into another.
In fact, the authors prove among other things, the following theorem: Let \(M\) and \(M'\) be real analytic hypersurfaces through the origin in \(\mathbb{C}^N\), \(N\geq 2\). Assume that neither \(M\) nor \(M'\) contains a nontrivial holomorphic subvariety through \(0\). Then any formal mapping \((\mathbb{C}^N, 0) \to(\mathbb{C}^N,0)\), mapping \(M\) into \(M'\) is analytic.
After giving the notion for \(M\) to be of finite type at \(0\), the authors prove the following: Let \(M\) and \(M'\) be real analytic generic submanifolds through \(0\) in \(\mathbb{C}^N\) with the same dimension. Assume that \(M\) is of finite type at \(0\), and does not contain any germ of nontrivial holomorphic subvariety through 0. Then \(M\) and \(M'\) are formally equivalent at 0 if and only if they are biholomorphically equivalent at 0.

MSC:
32V99 CR manifolds
32V40 Real submanifolds in complex manifolds
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
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