CR singular immersions of complex projective spaces
type: Item
instance of: scholarly article
title: CR singular immersions of complex projective spaces
zbMATH Open document ID: 1029.32020
author: Adam Coffman
published in: Beiträge zur Algebra und Geometrie
publication date: 2003-02-10T00:00:00Z
full work available at URL: https://eudml.org/doc/230474
full work available at URL: 13.html
review text: 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.
reviewed by: Xianghong Gong
Mathematics Subject Classification ID: 32V40
Mathematics Subject Classification ID: 14P05
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 32Q40
Mathematics Subject Classification ID: 15A22
Mathematics Subject Classification ID: 32S20
zbMATH DE Number: 1866004
zbMATH Keywords: CR singular immersions
zbMATH Keywords: complex projective spaces
zbMATH Keywords: rational mappings
zbMATH Keywords: equivalence
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1866004