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Chern-Moser operators and weighted jet determination problems. (English) Zbl 1232.32024
Barkatou, Y. (ed.) et al., Geometric analysis of several complex variables and related topics. Proceedings of the workshop, Marrakesh, Morocco, May 10–14, 2010. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-5257-6/pbk). Contemporary Mathematics 550, 75-88 (2011).
The authors use a generalization of the Chern-Moser operator to study local infinitesimal automorphisms of degenerate real-analytic hypersurfaces. In particular, they prove that the infinitesimal automorphisms of holomorphically non-degenerate model hypersurfaces are polynomials of weighted degree \(\leq 1\). This generalises a well-known fact for quadrics. Furthermore they prove that any infinitesimal automorphism of a hypersurface of finite type is completely determined by its weighted \(\mu_0\)-jet, if it is known that the infinitesimal automorphisms of the corresponding model surface have at most weighted degree \(\mu_0\). The results are illustrated by the model surface
\[ M_H=\big\{(z_1,z_2,w)\in \mathbb C^3: \text{Im\,} w = z^{}_1\bar{z}_2^l+ z_2^l \bar{z}^{}_1, l>1\big\}. \]
For the entire collection see [Zbl 1221.32001].

32V40 Real submanifolds in complex manifolds
32V35 Finite-type conditions on CR manifolds