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Deformations of reductive group actions

Published online by Cambridge University Press:  24 October 2008

Gerd Müller
Gerd Müller
Affiliation:
Fachbereich Mathematik, Universität Mainz, Saarstraβe 21, D-6500 Mainz, West Germany
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Consider actions of a reductive complex Lie group G on an analytic space germ (X, 0). In a previous paper [16] we proved that such an action is determined uniquely (up to conjugation with an automorphism of (X, 0)) by the induced action of G on the tangent space of (X, 0). Here it will be shown that every deformation of such an action, parametrized holomorphically by a reduced analytic space germ, is trivial, i.e. can be obtained from the given action by conjugation with a family of automorphisms of (X, 0) depending holomorphically on the parameter. (For a more precise formulation in terms of actions on analytic ℂ-algebras, see Theorem 2 below. An analogue for actions on formal ℂ-algebras is given there too.)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 106 , Issue 1 , July 1989 , pp. 77 - 88
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

REFERENCES

[1]Bierstone, E. and Milman, P.. Invariant solutions of analytic equations. Enseign. Math. (2) 25 (1979), 115–130.Google Scholar
[2]Bourbaki, N.. Éléments de mathématique. Livre VI, Intégration, Chapitres 1, 2, 3 et 4. Actualités Scientifiques et Industrielles vol. 1175 (Hermann, 1965).Google Scholar
[3]Bungart, L.. Holomorphic functions with values in locally convex spaces and applications to integral formulas. Trans. Amer. Math. Soc. 111 (1964), 317–344.CrossRefGoogle Scholar
[4]Forster, O.. Funktionswerte als Randintegrale in komplexen Räumen. Math. Ann. 150 (1963), 317–324.CrossRefGoogle Scholar
[5]Galligo, A.. Théorème de division et stabilité en géométrie analytique locale. Ann. Inst. Fourier (Grenoble) 29 2 (1979), 107–184.CrossRefGoogle Scholar
[6]Grauert, H. and Remmert, R.. Komplexe Räume. Math. Ann. 136 (1958), 245–318.CrossRefGoogle Scholar
[7]Grauert, H. and Remmert, R.. Analytische Stellenalgebren. Grundlehren math. Wissenschaften vol. 176 (Springer-Verlag, 1971).CrossRefGoogle Scholar
[8]Hauser, H. and Müller, G.. Algebraic singularities have maximal reductive automorphism groups. Nagoya Math. J. 113 (1989), 181–186.CrossRefGoogle Scholar
[9]Hochschild, G. P.. Basic theory of algebraic groups and Lie algebras. Graduate Texts in Math. vol. 75 (Springer-Verlag, 1981).CrossRefGoogle Scholar
[10]Hochschild, G. and Mostow, G. D.. Representations and representative functions of Lie groups, III. Ann. of Math. (2) 70 (1959), 85–100.CrossRefGoogle Scholar
[11]Humphreys, J. E.. Linear algebraic groups. Graduate Texts in Math. vol. 21 (Springer-Verlag, 1975).CrossRefGoogle Scholar
[12]Jurchescu, M.. On the canonical topology of an analytic algebra and of an analytic module. Bull. Soc. Math. France 93 (1965), 129–153.CrossRefGoogle Scholar
[13]Kaup, W.. Reelle Transformationsgruppen und invariante Metriken auf komplexen Räumen. Invent. Math. 3 (1967), 43–70.CrossRefGoogle Scholar
[14]Kaup, W.. Einige Bemerkungen über Automorphismengruppen von Stellenringen. Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. 1967 (1968), 43–50.Google Scholar
[15]Koras, M.. Deformations of actions of linearly reductive groups. Ann. Soc. Math. Polon. Ser. I Comment. Math. 22 (1980), 81–83.Google Scholar
[16]Müller, G.. Reduktive Automorphismengruppen analytischer ℂ-Algebren. J. Heine Angew. Math. 364 (1986), 26–34.Google Scholar
[17]Müller, G.. Actions of complex Lie groups on analytic ℂ-algebras. Monatsh. Math. 103 (1987), 221–231.CrossRefGoogle Scholar
[18]Müller, G.. Remarks on iteration of formal automorphisms. Aequationes Math. 35 (1988), 17–22.CrossRefGoogle Scholar
[19]Richardson, R. W.. Deformations of Lie subgroups and the variation of isotropy subgroups. Ada Math. 129 (1972), 35–73.Google Scholar