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**Holomorphic slices, symplectic reduction and multiplicities of representations.**
*(English)*
Zbl 0827.32030

Let \(M\) be a (not necessarily compact) connected Kähler manifold on which the complexification \(G^\mathbb{C}\) of a connected compact real Lie group \(G\) acts holomorphically. The main result of this paper is a theorem asserting the existence of slices for the \(G^\mathbb{C}\)-action at points in the zero level set of some moment map. The proof is a nice use of Kähler geometry, including a result on the interpolation of Kähler metrics near a totally real submanifold and Hörmander’s \(L^2\)- estimates for the Cauchy-Riemann operator.

The author applies this holomorphic slice theorem to the study of symplectic quotients of a Kähler manifold \(M\) on which \(G^\mathbb{C}\) acts holomorphically. If \(\Phi\) is an equivariant moment map for the action of the real form \(G\) and \(\lambda \in {\mathfrak g}^*\), then the symplectic quotient of \(M\) at the level \(\lambda\) is the space \(M_\lambda = \Phi^{-1} (G \lambda)/G\), where \(G \lambda\) is the coadjoint orbit through \(\lambda\). The author shows that \(M_\lambda\) is an analytic space and that the stratification on it by \(G\)-orbit types is analytic. A well-known result of Kirwan and Ness states that the symplectic quotient \(M_0\) of \(M\) (which is now assumed to be integral) is the same as \(M^{ss}//G^\mathbb{C}\) (semistability is analytic there) if the Kodaira embedding into projective space preserve the symplectic form. The author generalizes this result to the case when the Kodaira embedding is not a symplectic embedding. Thus there are many inequivalent Kähler structures on the algebraic quotients of \(M\). A corollary of this generalizes the Guillemin-Sternberg multiplicity formula to the case when the symplectic quotient \(M_\lambda\) is singular.

The author applies this holomorphic slice theorem to the study of symplectic quotients of a Kähler manifold \(M\) on which \(G^\mathbb{C}\) acts holomorphically. If \(\Phi\) is an equivariant moment map for the action of the real form \(G\) and \(\lambda \in {\mathfrak g}^*\), then the symplectic quotient of \(M\) at the level \(\lambda\) is the space \(M_\lambda = \Phi^{-1} (G \lambda)/G\), where \(G \lambda\) is the coadjoint orbit through \(\lambda\). The author shows that \(M_\lambda\) is an analytic space and that the stratification on it by \(G\)-orbit types is analytic. A well-known result of Kirwan and Ness states that the symplectic quotient \(M_0\) of \(M\) (which is now assumed to be integral) is the same as \(M^{ss}//G^\mathbb{C}\) (semistability is analytic there) if the Kodaira embedding into projective space preserve the symplectic form. The author generalizes this result to the case when the Kodaira embedding is not a symplectic embedding. Thus there are many inequivalent Kähler structures on the algebraic quotients of \(M\). A corollary of this generalizes the Guillemin-Sternberg multiplicity formula to the case when the symplectic quotient \(M_\lambda\) is singular.

Reviewer: J.S.Joel (Kelly)

### MSC:

32M05 | Complex Lie groups, group actions on complex spaces |

22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

22E46 | Semisimple Lie groups and their representations |

53D50 | Geometric quantization |

32Q15 | Kähler manifolds |

14L24 | Geometric invariant theory |

14L30 | Group actions on varieties or schemes (quotients) |