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Reduction of complex Hamiltonian \(G\)-spaces. (English) Zbl 0816.53018

Let \((X,\omega)\) be a Kähler space, in the sense that \(X\) is a complex space with an open cover \((U_ \alpha)\) and \(\omega\) is a family of continuous strictly plurisubharmonic functions \(\varphi_ \alpha : U_ \alpha \to \mathbb{R}\) such that there are \(f_{\alpha \beta} \in {\mathcal O}(U_ \alpha\cap U_ \beta)\) whose real parts equal \(\varphi_ \alpha - \varphi_ \beta\) on \(U_ \alpha \cap U_ \beta\). Where \(X\) and the \(\varphi_ \alpha\) are smooth they define a Kähler form on \(X\) by \({i\over 2} \partial\overline{\partial} \varphi_ \alpha\). Suppose that \(G\) is a complex reductive Lie group acting holomorphically on \(X\) with a maximal compact subgroup \(K\) which preserves the Kähler form. Let \(\mu : X \to \text{Lie}^* K\) be a moment map, i.e. \(\mu\) is a (sufficiently smooth) \(K\)-equivariant map whose \(\xi\)-th coordinate for any \(\xi \in \text{Lie }K\) is a Hamiltonian function for the vector field given by the infinitesimal action of \(\xi\) on \(X\). The authors call such a \((X,\omega, G, \mu)\) a complex Hamiltonian \(G\)-space. They prove that \(R = \mu^{- 1}(0)\) has the following properties:
(a) for \(x \in X^ R = \{x \in X \mid \overline{Gx} \cap R \neq \emptyset\}\) the orbit \(Gx\) is closed in \(X^ R\) if and only if \(Gx \cap R \neq \emptyset\);
(b) \(Gx \cap R = Kx\) if \(x \in R\);
(c) there is a \(G\)-invariant holomorphic map \(\pi : X^ R \to X^ R_ 0\) such that (i) for any open subset \(U_ 0\) of \(X^ R_ 0\) the inclusion of \({\mathcal O}(U_ 0)\) in \({\mathcal O}(\pi^{-1} U_ 0)^ G\) given by \(f\mapsto \pi^* f\) is an isomorphism; (ii) every fibre \(\pi^{-1}(y_ 0)\) contains exactly one closed \(G\)-orbit; (iii) the inclusion of \(R\) in \(X^ R\) induces a homeomorphism from \(R/K\) to \(X^ R_ 0\). This generalizes earlier results of Kempf-Ness, Kirwan, Sjamaar and others in the case when \(X\) is smooth.
Reviewer: F.Kirwan (Oxford)

MSC:

53C10 \(G\)-structures
32U05 Plurisubharmonic functions and generalizations
32E10 Stein spaces
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References:

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