Mike Usher

Lagrangian submanifolds of $\mathbb CP^n$

Usher mostly describes constructions of Lagrangians. (Joint work with Joel Oakley.)

Some nice examples:

• Chekanov torus in $\mathbb C^2$ (comes in 1-parametric family by "radius"). Chekanov-Eliashberg-Polterovich: these are Lagrangian isotopic to the standard "Clifford" tori, but not Hamiltonian isotopic. (In particular, same Maslov and area classes.)
• Both of these can then be shoved in $\mathbb{C}P^2$ and in $S^2 \times S^2$, and are monotone for specific values of the parameter $r$. (Non-displaceable)
• Polterovich torus in $T^*S^2$. Albers and Frauenfelder: monotone & non-displaceable.
• Higher dimensional version $P^r_{k,m}$ $k \ge 0$, $m \ge 1$.  Take $S^k \times \{0\}$ and $\{ 0 \} \times S^m$ in $S^{k+m+1}$. Look at the union of lifts of geodesics of speed $r$ passing through these two sets. This is monotone, non-displaceable Lagrangian diffeo to $S^1 \times S^k \times S^{m}/ (-id, -id, -id)$ (Polterovich torus corresponds to taking $k=0$ and $m=1$.)
• Passing to a quotient, can get these to live in $T^* \mathbb{R}P^{k+m+1}$. Embedding these in $S^2 \times S^2$ or in $\mathbb CP^n$, get shove these in here.
• Biran construction: $\mathbb{C}P^n = \mathbb{R}P^n \cup (\text{disk bundle over }Q_n)$. 

The main theorem is an alternative construction of Chekanov in dimension 4 that generalizes differently. In (much?) higher dimensions, however, this construction generalizes to a surprising displaceable monotone Lagrangian.

In dimension 6, get Lagrangian torus and Lagrangian $S^1 \times S^2$ whose displaceability are unknown, but that have vanishing Floer homology over $\mathbb Z$.