×

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.

MSC:

32E10 Stein spaces
32U05 Plurisubharmonic functions and generalizations
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.