## 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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### References:

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