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CR singular immersions of complex projective spaces. (English) Zbl 1029.32020

Consider the rational mappings f from P 1 into P 2 of the form [z]PP 0 ([z]), where P 0 ([z 0 ,z 1 ])=[z 0 z ¯ 0 ,z 0 z ¯ 1 ,z 1 z ¯ 0 ,z 1 z ¯ 1 ] and P:P 3 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 2×2 complex matrix K, and shows that the equivalence of two such mappings implies the similarity of the corresponding 2×2 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 P 1 in P 2 with exactly two elliptic complex tangents, a continuous family of totally real immersions of P 1 in P 2 with one point of self-intersection, and finitely many others.

The author also studies the equivalence problem for analogous mappings from P 2 into 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.

MSC:
32V40Real submanifolds in complex manifolds
14P05Real algebraic sets
14E05Rational and birational maps
32Q40Embedding theorems
15A22Matrix pencils
32S20Global theory of singularities (analytic spaces)