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On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions. (English) Zbl 0803.32011

We prove that if \(M_ 1\) and \(M_ 2\) are algebraic real hypersurfaces in (possibly different) complex spaces of dimension at least two and if \(f\) is a holomorphic mapping defined near a neighborhood of \(M_ 1\) so that \(f(M_ 1) \subset M_ 2\), then \(f\) is also algebraic, i.e., the field generated by adding each element of \(f\) to the rational functions field is of finite extension. Results in this direction originate in the work of Poincaré; modern results were obtained by Webster in case \(M_ 1\) and \(M_ 2\) lie in the same complex space. When \(M_ 1\) and \(M_ 2\) are spheres, then the result is due to Webster, Faran, Cima-Suffridge, and Forstneric (in this specific case, \(f\) is rational, algebraic of degree 1).
The proof of our result is based on a careful analysis on the invariant varieties and reduces to the consideration of many cases. After a slight modification, the argument can also be used to prove a reflection principle, which answers a problem of Forstneric and which in turn allows our main result to be stated for mappings that are holomorphic on one side and \(C^{k+1}\) smooth up to \(M_ 1\) where \(k\) is the codimension.
Reviewer: X.Huang (St.Louis)

MSC:

32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
14E05 Rational and birational maps
14H05 Algebraic functions and function fields in algebraic geometry
32J99 Compact analytic spaces
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