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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.

##### MSC:
 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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