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CR singular immersions of complex projective spaces. (English) Zbl 1029.32020
Consider the rational mappings \(f\) from \(\mathbb{C} P^1\) into \(\mathbb{C} P^2\) of the form \([z]\mapsto P\circ P_0([z])\), where \(P_0([z_0,z_1])=[z_0\overline z_0, z_0\overline z_1, z_1\overline z_0, z_1\overline z_1]\) and \(P:\mathbb{C} P^3\to\mathbb{C} P^2\) is an onto complex linear map. Two such mappings \(f\) are equivalent, if they are equal after biholomorphic transformations of the domain and target, respectively.
The author gives a complete classification for the equivalence problem. For the proof, the author associates to each mapping \(f\) a \(2\times 2\) complex matrix \(K\), and shows that the equivalence of two such mappings implies the similarity of the corresponding \(2\times 2\) matrices under matrix transformations satisfying a reality condition. The converse is also true, which is however a consequence of the geometric interpretation of the normal form of the matrix \(K\). The complete set of equivalence classes of mappings \(f\) consists of a continuous family of embeddings of \(\mathbb{C} P^1\) in \(\mathbb{C} P^2\) with exactly two elliptic complex tangents, a continuous family of totally real immersions of \(\mathbb{C} P^1\) in \(\mathbb{C} P^2\) with one point of self-intersection, and finitely many others.
The author also studies the equivalence problem for analogous mappings from \(\mathbb{C} P^2\) into \(\mathbb{C} P^5\). The complete classification for this case and for the higher dimension case is unsettled in this paper. However, the author gives some interesting examples.

32V40 Real submanifolds in complex manifolds
14P05 Real algebraic sets
14E05 Rational and birational maps
32Q40 Embedding theorems for complex manifolds
15A22 Matrix pencils
32S20 Global theory of complex singularities; cohomological properties
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