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Remarks on the rigidity of CR-manifolds. (English) Zbl 1101.32018
Given a germ $$(M,p)$$ of real submanifold of $$\mathbb{C}^n$$, one can consider the local stability group $$\operatorname{Aut}(M,p)$$, consisting of the germs of local bi-holomorphic transformations of $$\mathbb{C}^n$$ that keep $$p$$ fixed and preserve the germ $$(M,p)$$. When $$(M,p)$$ is real-analytic, this group is known either to be a Lie group, when some non-degeneracy conditions are satisfied, or else infinite dimensional. The Authors show that the situation is quite different when $$(M,p)$$ is $$\mathcal{C}^\infty$$ smooth, but fails to be real-analytic. The main result is the following: assume that $$(M,p)$$ is smooth, has positive CR dimension and codimension, and that $$\operatorname{Aut}(M,p)$$ contains the union $$\mathbf{G}$$ of an increasing sequence of finite dimensional subgroups of $$\operatorname{Aut}(\mathbb{C}^n, p)$$. Then there is a new germ $$(\widetilde{M},p)$$, that is tangent to $$(M,p)$$ to infinite order, for which $$\operatorname{Aut}(\widetilde{M},p)=\mathbf{G}$$ and moreover all germs of local CR automorphisms of $$\widetilde{M}$$ at $$p$$, that leave $$p$$ fixed, are restrictions of germs in $$\mathbf{G}$$. Nice applications are the examples involving CR manifolds with an $$\mathbf{S}^1$$-action, where for instance one can take $$\mathbf{G}=\{\exp(2\pi i(\ell/2^m))\mid \ell,m\in\mathbb{N}\}$$.

##### MSC:
 32V40 Real submanifolds in complex manifolds 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
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