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Lectures on Holomorphic Curves in Symplectic and Contact Geometry (Work in Progress—Version 3.3)

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Lectures on Holomorphic Curves in Symplectic and Contact Geometry (Work in Progress—Version 3.3) Lectures on Holomorphic Curves in Symplectic and Contact Geometry (Work in progress—Version 3.3) May 7, 2015 Chris Wendl Department of Mathematics, University College London E-mail address: [email protected] c 2015 by Chris Wendl Paper or electronic copies for noncommercial use may be made freely without explicit permission from the author. All other rights reserved. Contents Preface v Version history ix A Note on Terminology xi Chapter 1. Introduction 1 1.1. Warm up: Holomorphic curves in Cn 1 1.2. Hamiltonian systems and symplectic manifolds 3 1.3. Some favorite examples 5 1.4. Darboux’stheoremandtheMoserdeformationtrick 7 1.5. From symplectic geometry to symplectic topology 10 1.6. Contact geometry and the Weinstein conjecture 13 1.7. Symplectic fillings of contact manifolds 19 Chapter 2. Fundamentals 25 2.1. Almost complex manifolds and J-holomorphic curves 25 2.2. Compatible and tame almost complex structures 29 2.3. Linear Cauchy-Riemann type operators 41 2.4. The linearization of ∂¯J and critical points 43 2.5. Review of distributions and Sobolev spaces 48 2.6. Linear elliptic regularity 55 2.7. Local existence of holomorphic sections 68 2.8. The similarity principle 69 2.9. Unique continuation 71 2.10. Intersections with holomorphic hypersurfaces 75 2.11. Nonlinear regularity 77 2.12. Some tools of global analysis 83 2.13. Local existence of J-holomorphic curves 88 2.14. A representation formula for intersections 94 2.15. Simple curves and multiple covers 102 2.16. Positivity of intersections 104 2.A. Appendix: Singular integral operators 110 2.B. Appendix: Elliptic operators in general 117 Chapter 3. Fredholm Theory 125 3.1. Some Banach spaces and manifolds 125 3.2. Formal adjoints 130 3.3. The Fredholm property 136 iii 3.4. The Riemann-Roch formula and transversality criteria 139 Chapter 4. Moduli Spaces 145 4.1. The moduli space of closed J-holomorphic curves 145 4.2. Classification of pointed Riemann surfaces 152 4.3. Fredholm regularity and the implicit function theorem 163 4.4. Transversality for generic J 170 4.5. Generic families 181 4.6. Transversality of the evaluation map 188 4.7. Generic J-holomorphic curves are immersed 191 Chapter 5. Bubbling and Nonsqueezing 197 5.1. Gromov’s nonsqueezing theorem 197 5.2. Monotonicity in the integrable case 205 5.3. Bubbling off 208 Bibliography 215 Preface The present book-in-progress began as a set of lecture notes written at a furious pace to accompany a graduate course on holomorphic curves that I taught at ETH Z¨urich in Spring 2009, and repeated at the Humboldt-Universit¨at zu Berlin in the 2009-10 Winter semester. In both iterations of the course, it quickly became clear that my conceived objectives for the notes were not really attainable within the length of the semester, but the project nonetheless took on a life of its own. I have written these notes with the following specific goals in mind: (1) To give a solid but readable presentation of the analytical foundations of closed holomorphic curves from a modern perspective; (2) To use the above foundation to explain a few of the classic applications to symplectic topology, such as Gromov’s nonsqueezing theorem [Gro85] and McDuff’s results on rational and ruled symplectic 4-manifolds [McD90]; (3) To use the aforementioned “modern perspective” to generalize everything as cleanly as possible to the case of punctured holomorphic curves, and then explain some applications to contact geometry such as the Weinstein conjecture [Hof93] and obstructions to symplectic fillings [Wen10b]. The choice of topics covered and their presentation is partly a function of my own preferences, as well as my perception of which gaps in the existing literature seemed most in need of filling. In particular, I have devoted special attention to a few topics that seem fundamental but are not covered in the standard book on this subject by McDuff and Salamon [MS04], e.g. the structure of Teichm¨uller space and of the moduli space of unparametrized holomorphic curves of arbitrary genus, existence results for local J-holomorphic curves, and regularity for moduli spaces with constrained derivatives. My choice of applications is biased toward those which I personally find the most beautiful and which admit proofs with a very geometric flavor. For most such results, there are important abstract invariants lurking in the background, but one need not develop them fully in order to understand the proofs, and for that reason I have left out topics such as gluing analysis and Gromov-Witten theory, on which I would in any case have nothing to add to the superb coverage in [MS04]. In order to save space and energy, I have also included nothing about holomorphic curves with boundary, but aimed to make up for this by devoting the last third of the book to punctured holomorphic curves, a topic on which there are still very few available expositions aimed at graduate students. My personal attitude toward technical details is essentially that of a non-analyst who finds analysis important: what this means is that I’ve tried very hard to create an accessible presentation that is as complete as possible without boring readers v who don’t enjoy analysis for its own sake. In contrast to [MS04], I have not put the discussion of elliptic regularity in an appendix but rather integrated it into the main exposition, where it is (I hope) less likely to be ignored. On the other hand, I have presented such details in less generality than would be theoretically possible, in most places only as much as seems essential for the geometric applications. One example of this is the discussion in Chapter 2 of a local representation formula that is both weaker and easier to prove than the famous result of Micallef and White [MW95], but still suffices for crucial applications such as positivity of intersections. If some hardcore analysts find this approach lazy, my hope is that at least as many hardcore topologists may benefit from it. About the current version. This book has been growing gradually for several years, and the current version contains a little over half of what I hope to include in the finished product: there is not yet any serious material on contact geometry (only a few main ideas sketched in the introduction), but the development of the technical apparatus for closed holomorphic curves is mostly complete. The main thing still missing from this technical development is Gromov’s compactness theorem, though a simple case of it is covered in Chapter 5 in order to prove the nonsqueezing theorem. I hope to add the chapter on Gromov compactness in the next major revision, along with further chapters covering the special analytical properties of closed holomorphic curves in dimension four, and applications to symplectic 4-manifolds. It should be mentioned that in the time since this project was begun, a substan- tial portion of the material that I eventually plan to include in later chapters has appeared in other (shorter) sets of lecture notes that were written for various mini- courses. In particular, a comprehensive exposition of my perspective on McDuff’s characterization of symplectic rational and ruled surfaces now appears in [Wena], and some of the extensions of these ideas to punctured holomorphic curves and contact 3-manifolds are covered in [Wenb]. Both are written with similar target audiences in mind and should be readable by anyone who has made it through the existing chapters of this book—in fact they assume less technical background, but provide brief reviews of analytical material that is treated here in much more detail. It remains a long-term goal that the main topics covered in [Wena, Wenb] should eventually be integrated into the present manuscript in some form. Acknowledgments. I’d like to thank a number of people who have contributed useful comments, ideas, explanations and encouragement on this project, including Peter Albers, Jonny Evans, Joel Fish, Paolo Ghiggini, Janko Latschev, Sam Lisi, Klaus Mohnke, and Dietmar Salamon. I would also like to thank Urs Fuchs for pointing out errors in the original version, and particular gratitude goes to Patrick Massot, who has recently been testing these notes on Master’s students at the Ecole´ Polytechnique and has suggested many valuable improvements as a result. A very large portion of what I know about this subject was originally imparted to me by Helmut Hofer, whose unpublished manuscript with Casim Abbas [AH] has also been an invaluable resource for me. Other invaluable resources worth mention- ing include of course [MS04], as well as the expository article [Sik94] by Sikorav. Most of the revision work for Version 3.3 was undertaken during a two-month research visit to the Ecole´ Polytechnique, and I would like to thank them for their hospitality. Request. As should by now be obvious, these notes are work in progress, and as such I welcome comments, questions, suggestions and corrections from anyone making the effort to read them. These may be sent to [email protected]. Version history Versions 1 and 2 of these notes were the versions written to accompany the lec- ture courses I gave at ETH and the HU Berlin in 2009 and 2010 respectively; both included preliminary versions of what are now Chapters 1 through 4, though those chapters have undergone considerable expansion since then. The first revision to appear on the arXiv at http://arxiv.org/abs/1011.1690 was Version 3.1 (No- vember 2010), which included the additional fifth chapter on Gromov’s nonsqueezing theorem. Here is an overview of what has been added since then.
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