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Kählerian reduction in steps. (English) Zbl 1203.53083

Campbell, H.E.A. (ed.) et al., Symmetry and spaces. In Honor of Gerry Schwarz on the occasion of his 60th birthday. Basel: Birkhäuser (ISBN 978-0-8176-4874-9/hbk; 978-0-8176-4875-6/ebook). Progress in Mathematics 278, 63-82 (2010).
The paper under review includes a nice discussion of the reduction theory centered on the situation of a Kähler manifold \(X\) that is acted on in a Hamiltonian fashion by a compact Lie group \(K\) such that the corresponding action extends to the complexified Lie group \(K^{\mathbb C}\). The main result established in the paper is that if \(L\) is a closed normal subgroup of \(K\), then the reduction of \(X\) by \(K\) can be alternatively performed by the following two steps: Step 1. Firstly compute the Kählerian reduction of \(X\) by \(L\). Step 2. Then compute the reduction by \(K/L\) of the stratified Hamiltonian Kähler space obtained at Step 1. The various stratifications obtained in this process are carefully dealt with. The final section of the paper is devoted to the construction of quotients in the geometric invariant theory as an example of Kählerian reduction in steps which ends up with projective algebraic spaces.
For the entire collection see [Zbl 1182.14002].

MSC:

53D20 Momentum maps; symplectic reduction
32M05 Complex Lie groups, group actions on complex spaces
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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