Symplectic topology of projective space: Lagrangians, local systems and twistors
The Clifford Foliation.Abstract
In this thesis we study monotone Lagrangian submanifolds of $\mathbb{C}\mathbb{P}^n$. 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 $\mathbb{C}\mathbb{P}^n$ with minimal Maslov number $n + 1$ must be homotopy equivalent to $\mathbb{R}\mathbb{P}^n$ (this is joint work with Jack Smith). We also show that, if a monotone Lagrangian in $\mathbb{C}\mathbb{P}^3$ 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 $\mathbb{C}\mathbb{P}^{2n + 1}$ which project to maximal totally complex submanifolds of $\mathbb{H}\mathbb{P}^n$ under the twistor fibration. By applying the above topological restrictions to such Lagrangians, we show that the only embedded maximal Kähler submanifold of $\mathbb{H}\mathbb{P}^n$ is the totally geodesic $\mathbb{C}\mathbb{P}^n$ and that an embedded, non-orientable, superminimal surface in $S^4 = \mathbb{H}\mathbb{P}^1$ is congruent to the Veronese $\mathbb{R}\mathbb{P}^2$. 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 $L_{\Delta} \subseteq \mathbb{H}\mathbb{P}^3$ becomes wide in characteristic 2, which is known to be impossible to achieve with rank 1 local systems. We deduce that $L_{\Delta}$ and $\mathbb{R}\mathbb{P}^3$ cannot be disjoined by a Hamiltonian isotopy.
