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

Consider the rational mappings $f$ from $ℂ{P}^{1}$ into $ℂ{P}^{2}$ of the form $\left[z\right]↦P\circ {P}_{0}\left(\left[z\right]\right)$, where ${P}_{0}\left(\left[{z}_{0},{z}_{1}\right]\right)=\left[{z}_{0}{\overline{z}}_{0},{z}_{0}{\overline{z}}_{1},{z}_{1}{\overline{z}}_{0},{z}_{1}{\overline{z}}_{1}\right]$ and $P:ℂ{P}^{3}\to ℂ{P}^{2}$ 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 $2×2$ 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 $ℂ{P}^{1}$ in $ℂ{P}^{2}$ with exactly two elliptic complex tangents, a continuous family of totally real immersions of $ℂ{P}^{1}$ in $ℂ{P}^{2}$ with one point of self-intersection, and finitely many others.

The author also studies the equivalence problem for analogous mappings from $ℂ{P}^{2}$ into $ℂ{P}^{5}$. The complete classification for this case and for the higher dimension case is unsettled in this paper. However, the author gives some interesting examples.

##### MSC:
 32V40 Real submanifolds in complex manifolds 14P05 Real algebraic sets 14E05 Rational and birational maps 32Q40 Embedding theorems 15A22 Matrix pencils 32S20 Global theory of singularities (analytic spaces)