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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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