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Semistable quotients. (English) Zbl 0922.32017
Let \(G\) be a complex reductive Lie group, and \(X\) be a reduced complex space with holomorphic \(G\)-action. A complex space with a holomorphic map \(\pi:X\to Y\) is called a semistable quotient of \(X\) with respect to the \(G\)-action if (1) \(\pi\) is a \(G\)-invariant locally Stein map, and (2) \(O(Y)=\pi_* (O(X)^G)\), where \(O(X)\) (resp. \(O(X)^G))\) denotes the algebra of \((G\)-invariant) holomorphic functions on \(X\). If a semistable quotient exists, then it is unique up to biholomorphism and will be denoted by \(X\| G\).
The authors prove that the semistable quotient \(X\| G\) exists if and only if \(X\| T\) exists for some maximal algebraic torus \(T\) in \(G\). The authors also construct an example of \(X\) and \(G\) such that \(X\| G\) does not exist, but nevertheless \(X\| T_0\) exists for all 1-dimensional algebraic subgroups \(T_0\) of \(T\).

MSC:
32M05 Complex Lie groups, group actions on complex spaces
32L05 Holomorphic bundles and generalizations
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