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Softcover ISBN:  9780821842362 
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Softcover ISBN:  9780821842362 
Product Code:  CRMM/26.S 
List Price:  $115.00 
MAA Member Price:  $103.50 
AMS Member Price:  $92.00 
eBook ISBN:  9781470417727 
Product Code:  CRMM/26.E 
List Price:  $110.00 
MAA Member Price:  $99.00 
AMS Member Price:  $88.00 
Softcover ISBN:  9780821842362 
eBook ISBN:  9781470417727 
Product Code:  CRMM/26.S.B 
List Price:  $225.00 $170.00 
MAA Member Price:  $202.50 $153.00 
AMS Member Price:  $180.00 $136.00 

Book DetailsCRM Monograph SeriesVolume: 26; 2005; 82 ppMSC: Primary 53; Secondary 14
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 Morsetheoretic argument deriving the Klyachko inequalities and some of their generalizations. It reviews various infinitedimensional 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 potentialtheoretic 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 copublished with the Centre de recherches mathématiques.
ReadershipGraduate students and researchers interested in symplectic geometry, algebraic geometry, and geometric combinatorics.

Table of Contents

Chapters

Introduction

The convexity theorem for Hamiltonian $G$spaces

A constructive proof of the nonabelian convexity theorem

Some elementary examples of the convexity theorem

Kähler potentials and convexity

Applications of the convexity theorem


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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 Morsetheoretic argument deriving the Klyachko inequalities and some of their generalizations. It reviews various infinitedimensional 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 potentialtheoretic 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 copublished with the Centre de recherches mathématiques.
Graduate students and researchers interested in symplectic geometry, algebraic geometry, and geometric combinatorics.

Chapters

Introduction

The convexity theorem for Hamiltonian $G$spaces

A constructive proof of the nonabelian convexity theorem

Some elementary examples of the convexity theorem

Kähler potentials and convexity

Applications of the convexity theorem