Abstract
Let Ibea compact normal complex space on which C* acts 'in a nice manner'. We describe all invariant open subsets U of X such that the holomorphic map U -> U/C* of U onto the categorical quotient for the category of compact complex spaces, U/C*, is locally Stein. The description depends on a partial ordering of the fixed point components which arises from the Bialynicki-Birula decompositions of X.
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