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Automorphisms of nondegenerate CR quadrics and Siegel domains. Explicit description. (English) Zbl 1042.32014
Let $$z= (z^1,\dots, z^n)$$, $$w= (w^1,\dots, w^k)$$ be coordinates in $$\mathbb{C}^{n+k}$$, $$k\geq 1$$, and $\langle z,z\rangle= (\langle z,z\rangle^1,\dots, \langle z,z\rangle^k)$ be a $$\mathbb{C}^k$$-valued Hermitian form on $$\mathbb{C}^n$$. Let $$C$$ be the interior of the convex hull of $$\{\langle z,z\rangle: z\in\mathbb{C}^n\}$$ and suppose,that $$C$$ is an acute cone, i.e. $$C$$ does not contain any entire line. Let $$V\supset C$$ be an open acute cone in $$\mathbb{R}^k$$. The domain $$\Omega_V= \{(z,w)\in \mathbb{C}^{n+k}: \text{Im\,}w- \langle z,z\rangle\in V\}$$ is called a Siegel domain of the second kind associated with $$V$$, while the quadric $$Q= \{(z, w)\in \mathbb{C}^{n+k}: \text{Im\,}w= \langle z,z\rangle\}$$ forms the Silov boundary of $$\Omega_V$$.
After recalling the non-degenerateness of quadrics , the authors give an explicit formula for one-parameter groups of automorphisms of arbitrary nondegenerate quadrics and for the automorphisms of Siegel domains of second kind. The authors also introduce a family of $$k$$-dimensional chains, which are analogs of one-dimensional Chern-Moser chains for hyper-quadrics and clarify their structure, which is used for the proof of the main results.

##### MSC:
 32V20 Analysis on CR manifolds 32N05 General theory of automorphic functions of several complex variables
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##### References:
 [1] Belošapka, V.K. Finite-dimensionality of the automorphism group of a real-analytic surface (in Russian),Izv. Akad. Nauk SSSR Ser. Mat.,52(2), 437–442, (1988). English transl. inMath. USSR-Izv.,32, (1989). [2] Belošapka, V.K. A uniqueness theorem for automorphisms of a nondegenerate surface in the complex space (in Russian),Mat. Zametki,47(3), 17–22, (1990). English transl. inMath. Notes,47, (1990). [3] Ežov, V.V. and Schmalz, G. Holomorphic automorphisms of quadrics,Math. Z.,216, 453–470, (1994). · Zbl 0806.32007 · doi:10.1007/BF02572334 [4] Ežov, V.V. and Schmalz, G. A matrix Poincaré formula for holomorphic automorphisms of quadrics of higher codimension. Real associative quadrics,J. Geom. Analysis,7(4), 559–573, (1997). [5] Ežov, V.V. and Schmalz, G. Poincaré automorphisms for nondegenerate CR quadrics,Math. Annalen,298, 79–87, (1994). · Zbl 0847.32015 · doi:10.1007/BF01459726 [6] Ežov, V.V. and Schmalz, G. A simple proof of Belošapka’s theorem on the parametrization of the automorphism group of CR-manifolds,Mat. Zametki (Math. Notes),61(6), 939–942, (1997). [7] Henkin, G.M. and Tumanov, A.E. Local characterization of holomorphic automorphisms of Siegel domains,Funkt. Analysis,17(4), 49–61, (1983). [8] Kaup, W., Matsushima, Y., and Ochiai, T. On the automorphisms and equivalences of generalized Siegel domains,Am. J. Math.,92(2), 475–497, (1970). · Zbl 0198.42501 · doi:10.2307/2373335 [9] Palinčak, N. On quadrics of high codimension (in Russian),Mat. zametki,55(5), 110–115, (1994). [10] Poincaré, H. Les fonctions analytiques de deux variables et la représentation conforme,Rend. Circ. Math. Palermo, 185–220, (1907). · JFM 38.0459.02 [11] Pyatetskii-Shapiro.Automorphic Functions and Geometry of Classical Domains, Gordon and Breach, New York, 1969. [12] Rothaus, O. Automorphisms of Siegel domains,Am. J. Math.,101(5), 1167–1179, (1979). · Zbl 0443.32022 · doi:10.2307/2374131 [13] Satake, I.Algebraic Structures of Symmetric Domains, Iwanami Shoten and Princeton University Press, 1980. · Zbl 0483.32017 [14] Tumanov, A.E. Finite dimensionality of the group of CR-automorphisms of a standard CR manifold and characteristic holomorphic mappings of Siegel domains (in Russian),USSR Izvestiya,32(3), 655–662, (1989). · Zbl 0666.32022 · doi:10.1070/IM1989v032n03ABEH000804
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