Germs of holomorphic mappings between real algebraic hypersurfaces

Mir, Nordine

Annales de l'Institut Fourier, Tome 48 (1998) no. 4, pp. 1025-1043.

Nous étudions les germes d’applications holomorphes entre hypersurfaces algébriques réelles de ${ℂ}^n$. Plus précisément, nous considérons deux germes d’hypersurfaces algébriques $(M,p_0)$ et $(M^{\prime },p_0^{\prime })$ dans ${ℂ}^n$, $n\ge 2$, et $H$: $({ℂ}^n,p_0)\to ({ℂ}^n,p_0^{\prime })$ une application holomorphe de rang générique maximal telle que $H\left(M\right)\subseteq M^{\prime }$ et $H(p_0)=p_0^{\prime }$. Nous montrons que si $M$ n’est pas Lévi-plate, alors la fonction dite de réflexion associée à $H$ est toujours algébrique. Par conséquent, si l’hypersurface cible est donnée sous une forme normale, la composante transverse de $H$ est algébrique (sans aucune autre hypothèse de non-dégénérescence sur les hypersurfaces). Une autre conséquence de notre résultat est le théorème bien connu de Baouendi et Rothschild qui affirme que tout biholomorphisme entre hypersurfaces algébriques réelles holomorphiquement non dégénérées de ${ℂ}^n$ est algébrique.

We study germs of holomorphic mappings between general algebraic hypersurfaces. Our main result is the following. If $(M,p_0)$ and $(M^{\prime },p_0^{\prime })$ are two germs of real algebraic hypersurfaces in ${ℂ}^{N+1}$, $N\ge 1$, $M$ is not Levi-flat and $H$ is a germ at $p_0$ of a holomorphic mapping such that $H\left(M\right)\subseteq M^{\prime }$ and $\mathrm{Jac}\left(H\right)\not\equiv 0$ then the so-called reflection function associated to $H$ is always holomorphic algebraic. As a consequence, we obtain that if $M^{\prime }$ is given in the so-called normal form, the transversal component of $H$ is always algebraic. Another corollary of our main result is that any biholomorphism between holomorphically nondegenerate algebraic hypersurfaces is always algebraic, a result which was previously proved by Baouendi and Rothschild.

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