Invariant plurisubharmonic exhaustions and retractions
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.
scientific article; zbMATH DE number 599661