Book contents
- Frontmatter
- Contents
- Preface
- 1 The Arnold-Liouville theorem
- 2 Lagrangian fibrations
- 3 Global action-angle coordinates and torus actions
- 4 Symplectic reduction
- 5 Visible Lagrangian submanifolds
- 6 Focus-focus singularities
- 7 Examples of focus-focus systems
- 8 Almost toric manifolds
- 9 Surgery
- 10 Elliptic and cusp singularities
- Appendix A Symplectic linear algebra
- Appendix B Lie derivatives
- Appendix C Complex projective spaces
- Appendix D Cotangent bundles
- Appendix E Moser’s argument
- Appendix F Toric varieties revisited
- Appendix G Visible contact hypersurfaces and Reeb flows
- Appendix H Tropical Lagrangian submanifolds
- Appendix I Markov triples
- Appendix J Open problems
- References
- Index
3 - Global action-angle coordinates and torus actions
Published online by Cambridge University Press: 06 July 2023
Book contents
- Frontmatter
- Contents
- Preface
- 1 The Arnold-Liouville theorem
- 2 Lagrangian fibrations
- 3 Global action-angle coordinates and torus actions
- 4 Symplectic reduction
- 5 Visible Lagrangian submanifolds
- 6 Focus-focus singularities
- 7 Examples of focus-focus systems
- 8 Almost toric manifolds
- 9 Surgery
- 10 Elliptic and cusp singularities
- Appendix A Symplectic linear algebra
- Appendix B Lie derivatives
- Appendix C Complex projective spaces
- Appendix D Cotangent bundles
- Appendix E Moser’s argument
- Appendix F Toric varieties revisited
- Appendix G Visible contact hypersurfaces and Reeb flows
- Appendix H Tropical Lagrangian submanifolds
- Appendix I Markov triples
- Appendix J Open problems
- References
- Index
Summary
We study the class of toric manifolds: integrable Hamiltonian systems with standard period lattices away from certain controlled singularities. We show that the boundary of the image of such an integrable system is piecewise linear and discuss the Atiyah–Guillemin–Sternberg convexity theorem and Delzant existence and uniqueness theorems. We illustrate all of this with many examples.
- Type
- Chapter
- Information
- Lectures on Lagrangian Torus Fibrations , pp. 31 - 44Publisher: Cambridge University PressPrint publication year: 2023