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Zbl 1029.32020
CR singular immersions of complex projective spaces.
(English)
[J] Beitr. Algebra Geom. 43, No.2, 451-477 (2002). ISSN 0138-4821; ISSN 2191-0383/e

Consider the rational mappings $f$ from $\bbfC P^1$ into $\bbfC P^2$ 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 $P:\bbfC P^3\to\bbfC P^2$ 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. \par 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 $\bbfC P^1$ in $\bbfC P^2$ with exactly two elliptic complex tangents, a continuous family of totally real immersions of $\bbfC P^1$ in $\bbfC P^2$ with one point of self-intersection, and finitely many others. \par The author also studies the equivalence problem for analogous mappings from $\bbfC P^2$ into $\bbfC P^5$. 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 2000:
*32V40 Real submanifolds in complex manifolds
14P05 Real algebraic sets
14E05 Birational correspondences
32Q40 Embedding theorems
15A22 Matrix pencils
32S20 Global theory of singularities (analytic spaces)

Keywords: CR singular immersions; complex projective spaces; rational mappings; equivalence

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