CR singular immersions of complex projective spaces.

*(English)*Zbl 1029.32020Consider 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.

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.

Reviewer: Xianghong Gong (Madison)