On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions

Huang, Xiaojun

Annales de l'Institut Fourier, Tome 44 (1994) no. 2, pp. 433-463.

Dans cet article nous montrons que si $M_1$ et $M_2$ sont des hypersurfaces algébriques réelles dans deux espaces complexes de dimension $\ge 2$ et si $f$ est une fonction holomorphe, définie dans un voisinage de $M_1$, de manière à ce que $f(M_1)\subset M_2$, alors $f$ est aussi algébrique. Notre preuve est basée sur une analyse précise des variétés invariantes et se réduit à la considération de nombreux cas. Après une légère modification, notre argument est aussi utilisé pour prouver un principe de réflexion, qui permet une extension de notre théorème principal aux fonctions holomorphes d’un côté de $M_1$ et de classe $C^{k+1}$, où $k$ est la codimension.

In this paper, we show 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. Our proof is based on a careful analysis on the invariant varieties and reduces to the consideration of many cases. After a slight modification, the argument is also used to prove a reflection principle, which 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.

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