Symplectic Geometry of Integrable Hamiltonian

Among all the Hamiltonian systems, the integrable ones have special geometric properties; in particular, their solutions are very regular and quasi-periodic. The quasi-periodicity of the solutions of an integrable system is a result of the fact that the system is invariant under a (semi-global) torus action. It is thus natural to investigate the symplectic manifolds that can be endowed with a (global) torus action. This leads to symplectic toric manifolds (Part B of this book). Physics makes a surprising come-back in Part A: to describe Mirror Symmetry, one looks for a special kind of Lagrangian submanifolds and integrable systems, the special Lagrangians. Furthermore, integrable Hamiltonian systems on punctured cotangent bundles are a starting point for the study of contact toric manifolds (Part C of this book).

Preface

ix

A Lagrangian Submanifolds

Michèle Audin

1

(84)

Introduction

3

(2)

I Lagrangian and special Lagrangian immersions in Cn

5

(44)

I.1 Symplectic form on Cn, symplectic vector spaces

5

(4)

I.1.a Symplectic vector spaces

5

(1)

I.1.b Symplectic bases

6

(1)

I.1.c The symplectic form as a differential form

7

(1)

I.1.d The symplectic group

8

(1)

I.1.e Orthogonality, isotropy

9

(1)

I.2 Lagrangian subspaces

9

(2)

I.2.a Definition of Lagrangian subspaces

9

(1)

I.2.b The symplectic reduction

10

(1)

I.3 The Lagrangian Grassmannian

11

(5)

I.3.a The Grassmannian An as a homogeneous space

11

(1)

I.3.b The manifold An

11

(2)

I.3.c The tautological vector bundle

13

(1)

I.3.d The tangent bundle to An

14

(1)

I.3.e The case of oriented Lagrangian subspaces

15

(1)

I.3.f The determinant and the Maslow class

15

(1)

I.4 Lagrangian submanifolds in Cn

16

(13)

I.4.a Lagrangian submanifolds described by functions

16

(4)

I.4.b Wave fronts

20

(5)

I.4.c Other examples

25

(3)

I.4.d The Gauss map

28

(1)

I.5 Special Lagrangian submanifolds in Cn

29

(10)

I.5.a Special Lagrangian subspaces

29

(3)

I.5.b Special Lagrangian submanifolds

32

(1)

I.5.c Graphs of forms

33

(2)

I.5.d Normal bundles of surfaces

35

(1)

I.5.e From integrate a systems

36

(2)

I.5.f Special Lagrangian submanifolds invariant under SO(n)

38

(1)

I.6 Appendices

39

(5)

I.6.a The topology of the symplectic group

39

(2)

I.6.b Complex structures

41

(1)

I.6.c Hamiltonian vector fields, integrable systems

41

(3)

Exercises

44

(5)

II Lagrangian and special Lagrangian submanifolds in symplectic and Calabi-Yau manifolds

49

(32)

II.1 Symplectic manifolds

49

(2)

II.2 Lagrangian submanifolds and immersions

51

(2)

II.2.a In cotangent bundles

51

(2)

II.3 Tubular neighborhoods of Lagrangian submanifolds

53

(7)

II.3.a Moser's method

53

(4)

II.3.b Tubular neighborhoods

57

(2)

II.3.c "Moduli space" of Lagrangian submanifolds

59

(1)

II.4 Calabi-Yau manifolds

60

(6)

II.4.a Definition of the Calabi-Yau manifolds

61

(1)

II.4.b Yau's theorem

62

(1)

II.4.c Examples of Calabi-Yau manifolds

63

(3)

II.4.d Special Lagrangian submanifolds

66

(1)

II.5 Special Lagrangians in real Calabi-Yau manifolds

66

(4)

II.5.a Real manifolds

66

(1)

II.5.b Real Calabi-Yau manifolds

67

(1)

II.5.c The example of elliptic curves

68

(1)

II.5.d Special Lagrangians in real Calabi-Yau manifolds

68

(2)

II.6 Moduli space of special Lagrangian submanifolds

70

(3)

II.7 Towards mirror symmetry?

73

(3)

II.7.a Fibrations in special Lagrangian submanifolds

74

(1)

II.7.b Mirror symmetry

75

(1)

Exercises

76

(5)

Bibliography

81

(4)

B Symplectic Toric Manifolds

Ana Cannas da Silva

85

(90)

Foreword

87

(2)

I Symplectic Viewpoint

89

(40)

I.1 Symplectic Toric Manifolds

89

(12)

I.1.1 Symplectic Manifolds

89

(1)

I.1.2 Hamiltonian Vector Fields

90

(1)

I.1.3 Integrable Systems

91

(3)

I.1.4 Hamiltonian Actions

94

(2)

I.1.5 Hamiltonian Torus Actions

96

(2)

I.1.6 Symplectic Toric Manifolds

98

(3)

I.2 Classification

101

(17)

I.2.1 Delzant's Theorem

101

(2)

I.2.2 Orbit Spaces

103

(1)

I.2.3 Symplectic Reduction

104

(2)

I.2.4 Extensions of Symplectic Reduction

106

(3)

I.2.5 Delzant's Construction

109

(4)

I.2.6 Idea Behind Delzant's Construction

113

(5)

I.3 Moment Polytopes

118

(11)

I.3.1 Equivariant Darboux Theorem

118

(1)

I.3.2 Morse Theory

119

(1)

I.3.3 Homology of Symplectic Toric Manifolds

120

(2)

I.3.4 Symplectic Blow-Up

122

(2)

I.3.5 Blow-Up of Toric Manifolds

124

(2)

I.3.6 Symplectic Cutting

126

(3)

II Algebraic Viewpoint

129

(36)

II.1 Toric Varieties

129

(11)

II.1.1 Affine Varieties

129

(2)

II.1.2 Rational Maps on Affine Varieties

131

(1)

II.1.3 Projective Varieties

132

(3)

II.1.4 Rational Maps on Projective Varieties

135

(1)

II.1.5 Quasiprojective Varieties

136

(1)

II.1.6 Toric Varieties

137

(3)

II.2 Classification

140

(14)

II.2.1 Spectra

140

(2)

II.2.2 Toric Varieties Associated to Semigroups

142

(1)

II.2.3 Classification of Affine Toric Varieties

143

(1)

II.2.4 Fans

144

(4)

II.2.5 Toric Varieties Associated to Fans

148

(5)

II.2.6 Classification of Normal Toric Varieties

153

(1)

II.3 Moment Polytopes

154

(25)

II.3.1 Equivariantly Projective Toric Varieties

154

(1)

II.3.2 Weight Polytopes

155

(1)

II.3.3 Orbit Decomposition

156

(2)

II.3.4 Fans from Polytopes

158

(2)

II.3.5 Classes of Toric Varieties

160

(1)

II.3.6 Symplectic vs. Algebraic

161

(4)

Bibliography

165

(4)

Index

169

(6)

C Geodesic Flows and Contact Toric Manifolds

Eugene Lerman

175

Foreword

177

(2)

I From toric integrable geodesic flows to contact toric manifolds

179

(14)

I.1 Introduction

179

(5)

I.2 Symplectic cones and contact manifolds

184

(9)

II Contact group actions and contact moment maps

193

(4)

III Proof of Theorem I.38

197

(24)

III.1 Homogeneous vector bundles and slices

198

(4)

III.2 The 3-dimensional case

202

(3)

III.3 Uniqueness of Symplectic toric manifolds

205

(8)

III.3.l Cech cohomology

211

(2)

III.4 Proof of Theorem I.38, part three

213

(8)

III.4.1 Morse theory on orbifolds

216

(5)

Appendix Hypersurfaces of contact type

221

(2)

Bibliography

223

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Symplectic Geometry of Integrable Hamiltonian Sytems

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