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Symplectic Topology of Projective Space: Lagrangians, Local Systems and Twistors

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Symplectic Topology of Projective Space: Lagrangians, Local Systems and Twistors Symplectic Topology of Projective Space: Lagrangians, Local Systems and Twistors Momchil Preslavov Konstantinov A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy of University College London. Department of Mathematics University College London March, 2019 2 I, Momchil Preslavov Konstantinov, confirm that the work presented in this thesis is my own, except for the content of section 3.1 which is in collaboration with Jack Smith. Where information has been derived from other sources, I confirm that this has been indicated in the work. 3 На семейството ми. Abstract n In this thesis we study monotone Lagrangian submanifolds of CP . Our results are roughly of two types: identifying restrictions on the topology of such submanifolds and proving that certain Lagrangians cannot be displaced by a Hamiltonian isotopy. The main tool we use is Floer cohomology with high rank local systems. We describe this theory in detail, paying particular attention to how Maslov 2 discs can obstruct the differential. We also introduce some natural unobstructed subcomplexes. We apply this theory to study the topology of Lagrangians in projective space. We prove that a n monotone Lagrangian in CP with minimal Maslov number n + 1 must be homotopy equivalent to n RP (this is joint work with Jack Smith). We also show that, if a monotone Lagrangian in CP3 has minimal Maslov number 2, then it is diffeomorphic to a spherical space form, one of two possible Euclidean manifolds or a principal circle bundle over an orientable surface. To prove this, we use algebraic properties of lifted Floer cohomology and an observation about the degree of maps between Seifert fibred 3-manifolds which may be of independent interest. n+ Finally, we study Lagrangians in CP2 1 which project to maximal totally complex subman- n ifolds of HP under the twistor fibration. By applying the above topological restrictions to such n Lagrangians, we show that the only embedded maximal Kähler submanifold of HP is the totally n geodesic CP and that an embedded, non-orientable, superminimal surface in S4 = HP1 is congruent to the Veronese RP2. Lastly, we prove some non-displaceability results for such Lagrangians. In particular, we show that, when equipped with a specific rank 2 local system, the Chiang Lagrangian 3 LD ⊆ CP becomes wide in characteristic 2, which is known to be impossible to achieve with rank 1 3 local systems. We deduce that LD and RP cannot be disjoined by a Hamiltonian isotopy. Impact Statement The research carried out in this thesis impacts several related areas of geometry. On the one hand, it contributes to its primary domain – symplectic topology – with new results, namely: a calculation which showcases a rarely used method (high rank local systems in monotone Floer theory) and two classification results in Lagrangian topology. One of these, done in collab- oration with Jack Smith, builds upon recent work of several other authors to give a satisfactory partial answer to a question asked in the field more than ten years ago. The other relies on results in low-dimensional topology and relates to the study of minimal surfaces in the four dimensional sphere. On the other hand, we give applications of these results to examples coming from algebraic and Riemannian geometry. In particular, our symplectic methods allow us to deduce some uniqueness results about Legendrian varieties. These varieties arise naturally in the study of quaternion-Kähler manifolds and are related to a famous open problem in Riemannian geometry – the LeBrun-Salamon conjecture. The relation of symplectic geometry to this problem has not been explored in the litera- ture and one may hope that it could lead to new insights. Acknowledgements First and foremost I would like to thank my supervisor Jonny Evans for asking me the questions which gave this thesis its beginning and for his unwavering support all the way to its completion. His enthusiasm, patience and interest in my work have been a constant driving force behind it and his encouragement and mentorship have been crucial at many stages throughout my doctoral studies. For all his support, I am truly grateful. I sincerely thank Jack Smith for sharing his knowledge with me and for the many stimulating and fun conversations which made our collaboration so enjoyable. Special thanks go also to my closest academic family – Emily Maw, Agustin Moreno, Tobias Sodoge and Brunella Torricelli – for always being there to exchange ideas, share frustration in confusion and make conference trips all the merrier. I am also indebted to the larger symplectic community in London, especially the organisers and participants in the “Symplectic Cut” seminar which offered a wonderfully stimulating learning environment. This thesis and my mathematical understanding have benefited from conversations with François Charette, Yankı Lekili, Stiven Sivek, Dmitry Tonkonog and Chris Wendl. I thank also my examiners Paul Biran and Jason Lotay for their careful reading and useful comments. Wolfram Mathematica was instrumental for numerical calculations and some key visualisations. My studies were funded by the Engineering and Physical Sciences Research Council [EP/L015234/1], the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London. It is a pleasure to be a part of this programme and to have shared the journey with the other students. My sincere thanks go out to them and to the people who envisioned, created and run the LSGNT (especially to the ever so reliable and considerate Nicky Townsend). I am also grateful to everyone at the UCL Mathematics department for providing such a welcoming and inspiring atmosphere. I would like to thank my flatmates and colleagues Antonio Cauchi and Kwok-Wing Tsoi (Gha- leo) for all the great moments we shared. I am also indebted to my friends – especially Ivan, Ivet, Kamen and Lyubcho – for being there for me and for tolerating my absence during the writing process. My family’s support has been invaluable to me. I could never thank them enough. Finally: thank you, Desi. Without you, I would not have made it. Contents 1 Introduction 11 1.1 History and context . 11 1.1.1 Origins . 11 1.1.2 Pseudoholomorphic curves and Lagrangian topology . 12 n 1.1.3 Monotone Lagrangians in CP ......................... 18 1.2 Overview of this thesis . 20 1.2.1 A motivating example . 20 1.2.2 Methods and results . 21 1.2.3 Structure of the thesis . 31 2 High rank local systems in monotone Floer theory 33 2.1 Preliminaries . 33 2.1.1 The Maslov class and monotonicity . 33 2.1.2 Local systems . 36 2.1.3 Pre-complexes . 39 2.2 Floer cohomology and local systems . 39 2.2.1 The obstruction section . 39 2.2.2 Definition, obstruction and invariance . 44 2.2.3 The monodromy Floer complex . 53 2.3 The monotone Fukaya category . 56 2.3.1 Setup . 56 2.3.2 Units and morphisms of local systems . 60 2.3.3 Closed-open string map and the AKS theorem . 63 2.3.4 Decomposing F(M) .............................. 66 2.3.5 Split-generation . 68 2.4 The pearl complex . 78 2.4.1 Definition and obstruction . 78 2.4.2 The spectral sequence and comparison with Morse cohomology . 85 Contents 8 2.4.3 The monodromy pearl complex . 87 2.4.4 Algebraic structures . 88 n 3 Topological restrictions on monotone Lagrangians in CP 91 n 3.1 Lagrangians which look like RP ............................ 96 3.2 Monotone Lagrangians in CP3 ............................. 100 3.2.1 Preliminaries on 3-manifolds . 101 3.2.2 Proof of the main result . 108 n+ n 4 Symplectic geometry of the twistor fibration CP2 1 ! HP 116 4.1 The Legendrian–Lagrangian correspondence . 117 4.1.1 Background . 117 4.1.2 Proof of the correspondence . 122 4.1.3 Known examples . 139 4.1.4 Type 1 twistor Lagrangians . 141 4.1.5 Type 2 twistor Lagrangians . 143 4.1.6 Legendrian curves in CP3 and the Chiang Lagrangian . 149 4.2 The Lagrangian equation for CP3 from a twistor perspective . 155 4.2.1 The general equation . 155 4.2.2 An example: the Clifford torus . 161 4.3 No vertical Hamiltonians . 164 5 Non-displaceability of some twistor Lagrangians 167 5.1 The Chiang Lagrangian and RP3 ............................ 167 5.1.1 Identifying Z1 and LD .............................. 167 5.1.2 Topology of LD ................................. 169 5.1.3 Computation of Floer cohomology with local coefficients . 174 5.1.4 Proof of non-displaceability . 184 5.1.5 Some additional calculations . 187 5.2 Orientable subadjoint Lagrangians . 188 Appendices 193 2 4 A Vertical gradient equation on L+S 193 B Indecomposable representations over F2 of the binary dihedral group of order 12 196 Bibliography 205 List of Figures 2.1 The structure maps . 58 2.2 Evaluating OC∗ on a Hochschild boundary . 71 4.1 The Clifford foliation . 164 5.1 The fundamental domain for LD............................. 171 5.2 A Morse function f : LD ! R.............................. 172 5.3 Another representation of the Morse function f ..................... 173 0 5.4 Parallel transport along g˜2................................ 175 ˜ 5.5 Parallel transport along d23................................ 175 5.6 Parallel transport along g˜3................................ 176 0 5.7 A pearly trajectory u = (u) connecting xi to x j..................... 179 5.8 Parametrising disc boundaries. 180 5.9 A lift of the path g0 .................................. 182 uB11 5.10 A lift of the path g1 .................................. 183 uB11 List of Tables n+ n 4.1 Twistor correspondences for submanifolds of CP2 1 and HP ............ 140 5.1 Parallel transport maps for the pearly trajectories. 183 Chapter 1 Introduction 1.1 History and context 1.1.1 Origins Symplectic geometry was born in the early 19th century through the works of Lagrange and Poisson on celestial mechanics (see [Mar09]).
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