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