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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.
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)
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