## 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

### Keywords:

retractions; plurisubharmonic exhaustion
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### References:

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