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Holomorphic maps of algebraic CR manifolds. (English) Zbl 0926.32044

The authors study algebraic properties of holomorphic maps between real algebraic manifolds in complex affine spaces (in general, of unequal dimension) from the point of view of commutative algebra. The main result is the following theorem: Let \(M\) be a real algebraic generic manifold in a neighborhood of a point \({\mathfrak p}\in M\) in \(\mathbb C^n (n > 1),\) and let \(_{\mathfrak p}\mathbf f\) be a germ of a holomorphic map at \(\mathfrak p\) taking \(M\) into a real algebraic set \(M'\) in \(\mathbb C^{n'}.\) Suppose that \(M\) is Segre-transversal at \(\mathfrak p.\) Then there exist open connected neighborhoods \(U\) of \(\mathfrak p\) in \(\mathbb C^n\) and \(U'\) of \(_{\mathfrak p}{\mathbf f}({\mathfrak p})\) in \(\mathbb C^{n'}\) and a representative map \(\mathfrak f\) holomorphic on \(U\) such that \({\mathfrak f}(M\cap U)\in M',\) and the following property holds: For every point \(\mathfrak a\) of an open dense subset of \(M\cap U,\) there exists a complex \(m\)-dimensional algebraic variety \(X_{\mathfrak a} \subset\mathbb C^{n'}\) such that \({\mathfrak f}({\mathfrak a}) \in X_{\mathfrak a}\) and \(X_{\mathfrak a}\cap U'\subset M',\) where \(m\) is the transcendence degree of \(_{\mathfrak p}{\mathbf f}.\)

MSC:

32V20 Analysis on CR manifolds
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32S65 Singularities of holomorphic vector fields and foliations
12F05 Algebraic field extensions
13B02 Extension theory of commutative rings
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