Convexity Properties of Hamiltonian Group Actions

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Brand new. We distribute directly for the publisher. 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 subgroup of a complex reductive group acting on a Khler manifold, based on potential-theoretic methods in several complex variables.This volume is recommended for independent study and is suitable for graduate students and researchers interested in symplectic geometry, algebraic geometry, and geometric combinatorics. Read more Show Less

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Overview

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 subgroup of a complex reductive group acting on a Kahler manifold, based on potential-theoretic methods in several complex variables.

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Product Details

  • ISBN-13: 9780821842362
  • Publisher: American Mathematical Society
  • Publication date: 9/8/2006
  • Series: CRMM , #26
  • Pages: 82
  • Product dimensions: 6.90 (w) x 9.90 (h) x 0.20 (d)

Table of Contents

Ch. 1 The convexity theorem for Hamiltonian G-spaces 7
Ch. 2 A constructive proof of the non-abelian convexity theorem 21
Ch. 3 Some elementary examples of the convexity theorem 31
Ch. 4 Kahler potentials and convexity 55
Ch. 5 Applications of the convexity theorem 71
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