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Convexity Properties of Hamiltonian Group Actions
Victor Guillemin, Massachusetts Institute of Technology, Cambridge, MA, and Reyer Sjamaar, Cornell University, Ithaca, NY
A co-publication of the AMS and Centre de Recherches Mathématiques.
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CRM Monograph Series
2005; 82 pp; softcover
Volume: 26
ISBN-10: 0-8218-4236-6
ISBN-13: 978-0-8218-4236-2
List Price: US$36
Member Price: US$28.80
Order Code: CRMM/26.S
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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 Kähler 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.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

Readership

Graduate students and researchers interested in symplectic geometry, algebraic geometry, and geometric combinatorics.

Reviews

"This monograph gives a beautiful account of a fundamental theme in symplectic geometry: convexity of moment map images ... Guillemin and Sjamaar give a succinct survey of convexity results in symplectic geometry. It is an excellent starting point for someone new to the field, as well as a useful reference for the experts."

-- Mathematical Reviews

Table of 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
  • Kähler potentials and convexity
  • Applications of the convexity theorem
  • Bibliography
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