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On the automorphism groups of algebraic bounded domains. (English) Zbl 0823.14005
Let \(D\) be a bounded domain in \(\mathbb{C}^ n\). By the theorem of H. Cartan, the group \(\operatorname{Aut} (D)\) of all biholomorphic automorphisms of \(D\) has a unique structure of a real Lie group such that the action \(\operatorname{Aut} (D)\times D\to D\) is real analytic. This structure is defined by the embedding \(C_ v: \operatorname{Aut} (D) \hookrightarrow D\times \text{Gl}_ n (\mathbb{C})\), \(f\mapsto (f(v), f_{*v})\), where \(v\in D\) is arbitrary. Here we restrict our attention to the class of domains \(D\) defined by finitely many polynomial inequalities. The appropriate category for studying automorphism of such domains is the Nash category. Therefore we consider the subgroup \(\operatorname{Aut}_ a (D) \subset \operatorname{Aut} (D)\) of all algebraic biholomorphic automorphisms which in many cases coincides with \(\operatorname{Aut} (D)\). Assume that \(n>1\) and \(D\) has a boundary point where the Levi form is non-degenerate. Our main result is that the group \(\operatorname{Aut}_ a (D)\) carries a unique structure of an affine Nash group such that the action \(\operatorname{Aut}_ a (D)\times D\to D\) is Nash. This structure is defined by the embedding \(C_ v: \operatorname{Aut}_ a (D) \hookrightarrow D\times \text{Gl}_ n (\mathbb{C})\) and is independent of the choice of \(v\in D\).
Reviewer: D.Zaitsev (Bochum)

MSC:
14H37 Automorphisms of curves
14P20 Nash functions and manifolds
32M05 Complex Lie groups, group actions on complex spaces
14P10 Semialgebraic sets and related spaces
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References:
[1] R. Benedetti and J.-J. Risler. Real algebraic and semi-algebraic sets. Actualites Mathematiques. Hermann Editeurs des Sciences et des Arts, 1990 · Zbl 0694.14006
[2] H. Cartan. Sur les groupes de transformations analytiques. Act. Sc. et Int., Hermann, Paris, 1935 · JFM 61.0370.02
[3] K. Diederich and J.E. Forn?ss. Applications holomorphes propre entre domaines ? bord analytique r?el. C.R. Acad. Sci. Paris,307(I): 321-324, 1988
[4] K. Diederich and S Webster. A reflection principle for degenerate real hypersurfaces. Duke math. journal,47(4): 835-843, 1980 · Zbl 0451.32008 · doi:10.1215/S0012-7094-80-04749-3
[5] S. Kaneyuki. On the automorphism groups of homogeneous bounded domains. J. Fac. Sci. Univ. Tokyo, (14): 89-130, 1967 · Zbl 0165.40402
[6] J.J. Madden and C.M. Stanton. One-dimensional Nash groups. Pacific Journal of Math.,154(2): 331-344, 1992 · Zbl 0723.14042
[7] D. Mumford. The red book of varieties and schemes, volume 1358 of Lect. Notes in Math. Springer, 1980
[8] R. Narasimhan. Several complex variables. Chicago Lectures in Mathematics. Univ. of Chicago Press, 1971 · Zbl 0223.32001
[9] S.I. Pin?uk. On the analytic continuation of holomorphic mappings. Math USSR Sbornik,27(3): 345-392, 1975
[10] R. Remmert and K. Stein. Eigentliche holomorphe Abbildungen. Math. Zeitschr.,73: 159-189, 1960 · Zbl 0102.06903 · doi:10.1007/BF01162476
[11] O. Rothaus. The construction of homogeneous convex cones. Bull. Amer. Math. Soc., (69): 248-250, 1963 · Zbl 0117.39501 · doi:10.1090/S0002-9904-1963-10938-6
[12] E.B. Vinberg, S.G. Gindikin, and I.I. Pyatetskii-Shapiro Classification and canonical realization of complex bounded homogeneous domains. Trans. Moscow. Math. Soc., (12): 404-437, 1963 · Zbl 0137.05603
[13] S. Webster. On the mapping problem for algebraic real hypersurfaces. Inventiones math.,43: 53-68, 1977 · Zbl 0355.32026 · doi:10.1007/BF01390203
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