On CR mappings of real quadric manifolds. (English) Zbl 0821.32015

Sukhov, Alexandr

Mich. Math. J. 41, No. 1, 143-150 (1994).

A \(CR\)-quadric \(M\) of real codimension \(d\) in \(\mathbb{C}^ n = \mathbb{C}^ k \times \mathbb{C}^ d = \{(z,w)\}\) is defined by equations of the form \(\text{Re} w_ j = \langle L_ j (z), \overline z \rangle\), \(j = 1, \dots, d\); here the \(L_ j\) are \(\mathbb{C}\)-linear maps \(\mathbb{C}^ k \to \mathbb{C}^ k\), and \(\langle, \rangle\) is the standard bilinear form on \(\mathbb{C}^ k\). Then \(M\) is a \(CR\)-manifold of \(CR\)-dimension \(k\). The Levi cone of \(M\) (in \(\mathbb{R}^ d)\) is the convex hull of the locus \(\{(\langle L_ 1 (z), \overline z \rangle, \cdots, \langle Ld(z), \overline z \rangle) : z \in \mathbb{C}^ k \}\).
Theorem: Assume that the Levi cone of \(M\) has nonempty interior. Suppose \(M'\) is another \(CR\)-quadric (in \(\mathbb{C}^{n'} = \mathbb{C}^{k'} \times \mathbb{C}^{d'})\) and \(F : D \to M'\) a \(CR\)-map of class \(C^ 1\) from a nonempty connected open subset \(D\) of \(M\). Then \(F\) extends to a complex rational map \(\mathbb{C}^ n \to \mathbb{C}^{n'}\), provided the following condition holds: assuming (without loss of generality since \(M\), \(M'\) are complex-affinely homogeneous) that \(O \in D\), and \(F(O) = 0\), we must have \(\sum^{d'}_{j = 1} L_ j'(dF(O) (\mathbb{C}^ k)) - \mathbb{C}^{k'}\) (note that \(\mathbb{C}^ k\) is the complex tangent space to \(M\) at \(O)\).
This theorem generalises Alexander’s famous theorem on local \(CR\)- diffeomorphisms of the sphere in \(\mathbb{C}^ n(n > 1)\), as well as its generalisations by Tumanov, Henkin and Forstneric.

Reviewer: R.R.Simha (Bombay)

MSC:

32V05	CR structures, CR operators, and generalizations
32V40	Real submanifolds in complex manifolds

Keywords:

\(CR\) mappings; \(CR\)-quadric; Levi cone