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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.

(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)

### Keywords:

Kähler space; Kähler form; reductive Lie group; Hamiltonian function; Hamiltonian \(G\)-space
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\textit{P. Heinzner} and \textit{F. Loose}, Geom. Funct. Anal. 4, No. 3, 288--297 (1994; Zbl 0816.53018)

### References:

[1] | [H]P. Heinzner, Geometric invariant theory on Stein spaces, Math. Ann. 289 (1991), 631–662. · Zbl 0728.32010 |

[2] | [HHuL]P. Heinzner, A.T. Huckleberry, F. Loose, Kählerian extension of the sympletic reduction, to appear in J. reine u. angew. Math. · Zbl 0803.53042 |

[3] | [KN]G. Kempf, L. Ness, The length of vectors in representation spaces, in ”Algebraic Geometry”, Lecture Notes in Mathematics 732, Springer (1979), 233–244. · Zbl 0407.22012 |

[4] | [Ki]F.C. Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry, Princeton University Press, 1984. · Zbl 0553.14020 |

[5] | [S]R. Sjamaar, Holomorphic slices, symplectic reduction and multiplicities of representations, Preprint MIT 1993. · Zbl 0827.32030 |

[6] | [Sc]G.W. Schwarz, The topology of algebraic quotients, in ”Topological Methods in Algebraic Transformation Groups”, Progress in Mathematics 80, Birkhäuser (1989), 135–151. |

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