Abstract
LetG be a complex reductive Lie group with maximal compact subgroupK andG×X →X a holomorphic action on a Stein manifoldX. LetRo andR1 be two Kempf-Ness sets arising from moment maps induced by strictly plurisubharmonic,K-invariant, proper functions. Then there is a globalK-equivariant diffeomorphism φ:X→X with φ(R0)=R1. In particular, the induced differentiable structures on the categorical quotientX G are diffeomorphic. The proof is based on a variant of Moser's method using time-dependent vector fields. An example shows that the differentiable structures can indeed be different, even though they are isomorphic.
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