Algebraicity of holomorphic mappings between real algebraic sets in \(\mathbb{C}^ n\) (Q1373010)

The main results of this extensive and substantial paper are related to the question under which conditions a germ \(H\) of a holomorphic map in \(\mathbb{C}^N\), mapping an irreducible real algebraic set \(A\subset \mathbb{C}^N\) into another real algebraic set of the same dimension, is actually algebraic. The paper contains a series of general results from which special results can be deduced: For example, if \(H\) is a germ as above in a regular point of \(A\) with Jac \(H\not \equiv 0\), then \(H\) is algebraic if the following sufficient conditions, which are also essentially necessary, are fulfilled: 1) \(A\) is holomorphically nondegenerate everywhere in some nonempty relatively open subset of the set of regular points of \(A\) and 2) Every holomorphic algebraic function \(f\) in a neighbourhood in \(\mathbb{C}^N\) of a point of \(A\), with \(f_{|A}\) real-valued, is constant. The authors prove that these conditions together are equivalent to the existence of the following three types of CR points in \(A\): holomorphically nondegenerate, minimal and generic ones. Several results are related to the case of real algebraic CR submanifolds \(M\) of \(\mathbb{C}^N\). It is shown that the CR orbit of every \(p\in M\) is a real algebraic submanifold of \(M\), and the intrinsic complexification \(X\) of this orbit is a complex algebraic submanifold of \(\mathbb{C}^N\). If \(H\) is the germ of a biholomorphic map in a neighbourhood in \(\mathbb{C}^N\) of \(p\in M\), mapping \(M\) into another real algebraic manifold of the same dimension, then the restriction of \(H\) to the complex algebraic manifold \(X\) is algebraic. Remarkably, the algebraicity results in this paper are deduced from analyticity properties of locally defined objects. The so-called Segre sets play a central methodical role. These are sequences of sets which are attached to the points of a real analytic CR manifold. The concept of these sets is described in detail and generalizes the concept of the so-called Segre surfaces which has been used before by other authors. The paper also contains a brief historical survey about previous work on the algebraicity of holomorphic mappings between real algebraic sets in \(\mathbb{C}^N\).