On the local equivalence of homogeneous CR-manifolds.

*(English)*Zbl 1092.32021Let us denote by \(\mathcal A\) the class of all CR manifolds \(M\) satisfying the following three conditions: a) \(M\) is compact and simply connected; b) it is locally homogeneous; c) there exists at least one point \(x \in M\) such that the Lie algebra of germs at \(x\) of the infinitesimal CR transformations of \(M\) is finite dimensional.

In this paper, the author shows that if \(M, M' \in \mathcal A\), then any local CR equivalence \(f: U \subset M \to U' \subset M'\) extends to a unique global CR equivalence \(f: M \to M'\).

The following application is given. In [Invent. Math. 153, No. 1, 45–104 (2003; Zbl 1027.32032)], the author and D. Zaitsev considered a special class of homogeneous CR manifolds, which are all realized as compact orbits in a complex vector space of linear actions of certain semisimple Lie groups. In that paper, the classification of those orbits up to global CR equivalence was given. With the result of this paper, the author can prove that the previous classification gives also the classification up to local CR equivalence of the orbits which are simply connected and of those which are non-simply connected but belong to a certain special subclass.

In this paper, the author shows that if \(M, M' \in \mathcal A\), then any local CR equivalence \(f: U \subset M \to U' \subset M'\) extends to a unique global CR equivalence \(f: M \to M'\).

The following application is given. In [Invent. Math. 153, No. 1, 45–104 (2003; Zbl 1027.32032)], the author and D. Zaitsev considered a special class of homogeneous CR manifolds, which are all realized as compact orbits in a complex vector space of linear actions of certain semisimple Lie groups. In that paper, the classification of those orbits up to global CR equivalence was given. With the result of this paper, the author can prove that the previous classification gives also the classification up to local CR equivalence of the orbits which are simply connected and of those which are non-simply connected but belong to a certain special subclass.

Reviewer: Andrea Spiro (Camerino)

##### MSC:

32V20 | Analysis on CR manifolds |

22F30 | Homogeneous spaces |

57S25 | Groups acting on specific manifolds |

32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |