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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
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