Invariant plurisubharmonic exhaustions and retractions. (English) Zbl 0842.32010

Let \(X\) be a Stein manifold equipped with an \(R\)-analytic \(K \times X \to X\) of a compact Lie group \(K\) of holomorphic transformations. There is a \(K\)-invariant strictly plurisubharmonic exhaustion \(\phi\): \(X \to \mathbb{R}^+\) which plays the role of a norm. The Kempf-Ness set \(R = R(\phi)\) can be defined as the zero-set of the associated moment norm-function and the flow is associated to the Kähler-form \(i \ni \overline{\delta} \phi\). The authors modify \(\phi\) outside of its Kempf-Ness set \(R = R(\phi)\) so that the resulting gradient flow realizes \(R(\phi)\) as a strong deformation retract of \(X\). As a consequence, they derive topological results which are analogous to those in the algebraic case and apply to a local existence for equivalent structures on topological \(K\)-bundles, which is a necessary tool for the proof of the equivalent version of Oka-Grauert’s principle. They also apply to a homotopically equivalent theorem of \(X\) and a universal Stein manifold \(X^c\) containing \(X\) as an open, Runge \(K\)-invariant subset.


32E10 Stein spaces
32U05 Plurisubharmonic functions and generalizations
Full Text: DOI EuDML


[1] [G] Grauert, H.: On Levi’s problem and the imbedding of real- analytic manifolds Ann. of Math.68, 460–473 (1958) · Zbl 0108.07804
[2] [H1] Heinzner, P.: Geometric invariant theory on Stein spaces. Math. Ann.289, 631–662 (1991) · Zbl 0728.32010
[3] [H2] Heinzner, P.: Equivariant holomorphic extensions of real analytic manifolds. To appear in Bull. Soc. Math. de France (1993) · Zbl 0794.32022
[4] [HK1] Heinzner, P.; Kutzschebauch, F.: Le principe d’Oka équivariant. C.R. Acad. Sci. Paris.315, 1265–1267 (1992) · Zbl 0782.32021
[5] [HK2] Heinzner, P.; Kutzschebauch, F.: Equivariant Oka-principle. Preprint-Bochum (1993) · Zbl 0837.32004
[6] [K] Kirwan, F.: Cohomology of quotients in symplectic and algebraic geometry. Mathematical notes 31, Princeton University Press, Princeton New Jersey, 1984 · Zbl 0553.14020
[7] [KN] Kempf, G.; Ness, L.: The length of vectors in representation spaces. Lect. Notes Math. vol.732, Springer-Verlag, Berlin Heidelberg New York, 233–243, 1979 · Zbl 0407.22012
[8] [Kr] Kraft, H.: Geometrische Methoden in der Invariantentheorie. Vieweg-Verlag, Braunschweig, 1984 · Zbl 0569.14003
[9] [KPR] Kraft, H.; Petrie, T.; Randall, J.D.: Quotient Varieties. Adv. Math.74, 145–162 (1989) · Zbl 0691.14029
[10] [N] Narasimhan, R.: Analysis on real and complex manifolds. Adv. Studies in pure math., North Holland, 1968
[11] [Ne] Neeman, A.: The topology of quotient varieties. Ann. of Math.103, 419–459 (1985) · Zbl 0692.14032
[12] [Sch] Schwarz, G. W.: The topology of algebraic quotients. Proceedings of the Rutgers Conference on Transformation Groups, 135–151 (1990)
[13] [S] Snow, D. M.: Reductive group actions on Stein Spaces. Math. Ann.259, 79–97 (1982) · Zbl 0509.32021
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