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
7 - Examples of focus-focus systems
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 construct examples of focus–focus systems which occur naturally on Milnor fibres of surface singularities. This allows us to read off aspects of the geometry and topology of these Milnor fibres. We also give a detailed analysis of Auroux’s integrable system on complex 2–space, which is a local model for understanding focus–focus fibrations.
- Type
- Chapter
- Information
- Lectures on Lagrangian Torus Fibrations , pp. 86 - 99Publisher: Cambridge University PressPrint publication year: 2023