Konstantinov, Momchil Preslavov; (2019) Symplectic Topology of Projective Space: Lagrangians, Local Systems and Twistors. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

In this thesis we study monotone Lagrangian submanifolds of CPn . 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 monotone Lagrangian in CPn with minimal Maslov number n + 1 must be homotopy equivalent to RPn (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. Finally, we study Lagrangians in CP(2n+1) which project to maximal totally complex subman- ifolds of HPn under the twistor fibration. By applying the above topological restrictions to such Lagrangians, we show that the only embedded maximal Kähler submanifold of HPn is the totally geodesic CPn 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 L∆ ⊆ CP3 becomes wide in characteristic 2, which is known to be impossible to achieve with rank 1 local systems. We deduce that L∆ and RP3 cannot be disjoined by a Hamiltonian isotopy.

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