Momentum maps and reduction in algebraic geometry
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.
scientific article; zbMATH DE number 1308132
scientific article