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基本説明
Presents a simple local model for the moment polytope, valid in the "generic" case, and an elementary Morse-theoretic argument deriving the Klyachko inequalities and some of their generalizations.
Full Description
This is a monograph on convexity properties of moment mappings in symplectic geometry. The fundamental result in this subject is the Kirwan convexity theorem, which describes the image of a moment map in terms of linear inequalities. This theorem bears a close relationship to perplexing old puzzles from linear algebra, such as the Horn problem on sums of Hermitian matrices, on which considerable progress has been made in recent years following a breakthrough by Klyachko. The book presents a simple local model for the moment polytope, valid in the "generic" case, and an elementary Morse-theoretic argument deriving the Klyachko inequalities and some of their generalizations. It reviews various infinite-dimensional manifestations of moment convexity, such as the Kostant type theorems for orbits of a loop group (due to Atiyah and Pressley) or a symplectomorphism group (due to Bloch, Flaschka, and Ratiu). Finally, it gives an account of a new convexity theorem for moment map images of orbits of a Borel su.
This volume is recommended for independent study and is suitable for graduate students and researchers interested in symplectic geometry, algebraic geometry, and geometric combinatorics.
Contents
Introduction; The convexity theorem for Hamiltonian $G$-spaces; A constructive proof of the non-abelian convexity theorem; Some elementary examples of the convexity theorem; Kahler potentials and convexity; Applications of the convexity theorem; Bibliography
