## On $$CR$$-mappings between algebraic Cauchy-Riemann manifolds and separate algebraicity for holomorphic functions.(English)Zbl 0851.32017

The present paper extends a result of S. Webster [Invent. Math. 43, 53-68 (1977; Zbl 0355.32026)], which asserts to the effect that a local biholomorphism between two real algebraic Levi-non-degenerate hypersurfaces extends to an algebraic mapping to all of $$\mathbb{C}^n$$, to the case of CR mappings between real algebraic Cauchy-Riemann manifolds of higher codimensions. More precisely, first of all one defines a generic real algebraic manifold $$M$$ of codimension $$d \geq 1$$ in a domain $$\Omega \subset \mathbb{C}^n$$, by $$M = \{z \in \Omega : \rho_j(z, \overline{z}) = 0$$, $$j =1,\dots, d\}$$, where $$\rho_j$$ are real polynomials with $$\overline{\partial} \rho_1 \wedge \dots \wedge \overline{\partial}\rho_d \neq 0$$ in $$\Omega$$ and these are called defining functions. A corollary of the main result asserts that if $$F : M \to M'$$ is a CR diffeomorphism of class $$C^1$$ between two real algebraic manifolds in $$\mathbb{C}^n$$ with non-degenerate Levi forms and non-degenerate Levi cones. Then $$F$$ extends to an algebraic mapping on all $$\mathbb{C}^n$$. Here the Levi cone (at $$p$$) is defined as the convex hull of the set $$\{(\alpha_1,\dots, \alpha_d) \in \mathbb{R}^d$$: $$\alpha_j = \partial \overline {\partial}\rho_j (u,\overline{u})$$, $$u \in T^c_p(M)\}$$. The Levi cone (at $$p$$) is said to be non-degenerate (at $$p$$) if it has a non-empty interior in $$\mathbb{R}^d$$. The proof is based on a modified version of Webster reflection principle [the second author, Mich. Math. J. 41, No. 1, 143-150 (1994; Zbl 0821.32015)], and a generalization of the classical separate algebraicity theorem (if a function on a domain $$D \subset \mathbb{C}^n$$ is algebraic in each separate variable when others are fixed, then the function is an algebraic function on $$D$$), where the parallel lines are replaced by families of algebraic curves.

### MSC:

 32D15 Continuation of analytic objects in several complex variables 32D99 Analytic continuation 32H99 Holomorphic mappings and correspondences 32V40 Real submanifolds in complex manifolds

### Citations:

Zbl 0355.32026; Zbl 0821.32015
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### References:

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