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Momentum maps and reduction in algebraic geometry. (English) Zbl 0955.37031
The paper gives a brief survey of the works in the last fifteen years devoted to the study of the geometry of the momentum map with applications to symplectic geometry and to related topics in algebraic geometry. The problem of understanding how the geometry and topology of the symplectic reduction at a coadjoint orbit varies as the orbit varies and what happens when the symplectic reduction acquires singularities is discussed. This problem is explained to be related to the corresponding questions in algebraic geometry, namely, how the quotient varieties which arise in Mumford’s geometric invariant theory vary with the linearization of the group action and how their singularities appear. The relations of the problems mentioned above with the theory of moduli spaces of vector bundles, residue formulae and nonabelian localization problems and with the theory of Yang-Mills equations are also indicated.

MSC:
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
14D20 Algebraic moduli problems, moduli of vector bundles
14L30 Group actions on varieties or schemes (quotients)
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
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