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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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