Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Introduction: Motivation
- Lecture 1 Closed Holomorphic Curves in Symplectic 4-Manifolds
- Lecture 2 Intersections, Ruled Surfaces, and Contact Boundaries
- Lecture 3 Asymptotics of Punctured Holomorphic Curves
- Lecture 4 Intersection Theory for Punctured Holomorphic Curves
- Lecture 5 Symplectic Fillings of Planar Contact 3-Manifolds
- Appendix A Properties of Pseudoholomorphic Curves
- Appendix B Local Positivity of Intersections
- Appendix C A Quick Survey of Siefring’s Intersection Theory
- References
- Index
Lecture 1 - Closed Holomorphic Curves in Symplectic 4-Manifolds
Published online by Cambridge University Press: 06 March 2020
Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Introduction: Motivation
- Lecture 1 Closed Holomorphic Curves in Symplectic 4-Manifolds
- Lecture 2 Intersections, Ruled Surfaces, and Contact Boundaries
- Lecture 3 Asymptotics of Punctured Holomorphic Curves
- Lecture 4 Intersection Theory for Punctured Holomorphic Curves
- Lecture 5 Symplectic Fillings of Planar Contact 3-Manifolds
- Appendix A Properties of Pseudoholomorphic Curves
- Appendix B Local Positivity of Intersections
- Appendix C A Quick Survey of Siefring’s Intersection Theory
- References
- Index
Summary
The first lecture introduces the study of closed symplectic manifolds via closed pseudoholomorphic curves, focusing in particular on McDuff’s famous theorem from 1990 that characterizes symplectic ruled surfaces via symplectically embedded spheres. The theorem is stated in a slightly more modern formulation using Lefschetz fibrations, and a proof modulo of some technical lemmas is sketched. The topic of intersection theory is then introduced by considering the natural question of what kinds of holomorphic curves can be proven to form leaves of smooth foliations by holomorphic curves.
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- Publisher: Cambridge University PressPrint publication year: 2020