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Mappings of real algebraic hypersurfaces. (English) Zbl 0869.14025
A smooth real hypersurface in \(\mathbb{C}^N\) is called algebraic if it is contained in the zero set of a nonzero real-valued polynomial in \(Z\) and \(\overline Z\). A hypersurface \(M\) in \(\mathbb{C}^N\) is holomorphically degenerate at a point \(p_0 \in M\) if there exists a nonzero germ of a holomorphic vector field tangent to \(M\) in a neighborhood of \(p_0\). The paper contains a complete characterization of algebraic hypersurfaces in \(\mathbb{C}^N\) such that any holomorphic map with non-vanishing Jacobian determinant between two such hypersurfaces is algebraic. Let \(M\) be a connected algebraic hypersurface in \(\mathbb{C}^N\), \(N>1\), and let \(p_0\) be a point of \(M\). Every biholomorphism defined in a neighborhood of \(p_0\) and mapping \(M\) to another algebraic hypersurface in \(\mathbb{C}^N\) is algebraic if and only if \(M\) is not holomorphically degenerate at any point. This is a generalization of a result of S. M. Webster [Invent. Math. 43, 53-68 (1977; Zbl 0348.32005)].
The authors also describe a relationship between essential finiteness and holomorphic nondegeneracy. If \(M\) is a connected real analytic hypersurface in \(\mathbb{C}^N\), then there exists a point of \(M\) in which \(M\) is essentially finite if and only if \(M\) is not holomorphically degenerate at any point in \(M\).

MSC:
14P99 Real algebraic and real-analytic geometry
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
14J70 Hypersurfaces and algebraic geometry
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