Continuum families of non-displaceable Lagrangian tori in $(\mathbb{C}P^1)^{2m}$

TL;DR

This paper constructs and analyzes families of non-displaceable Lagrangian tori in complex projective spaces, demonstrating their superheaviness and non-displaceability properties, and extends results to certain symplectic forms on blow-ups of projective planes.

Contribution

It introduces new continuum families of non-displaceable Lagrangian tori in $(C P^1)^{2m}$ and establishes their superheaviness and non-displaceability properties, extending known constructions.

Findings

Constructed families of non-displaceable Lagrangian tori in $(C P^1)^n$.

Proved superheaviness of certain tori with respect to partial symplectic quasi-states.

Extended results to symplectic forms on $C P^2 ar{C P}^2$.

Abstract

We construct a family of Lagrangian tori $\Theta^n_s$ $\subset$ $(\mathbb{C}P^1)^n$, $s \in (0,1)$, where $\Theta^n_{1/2} = \Theta^n$, is the monotone twist Lagrangian torus described by Chekanov-Schlenk. We show that for $n = 2m$ and $s \ge 1/2$ these tori are non-displaceable. Then by considering $\Theta^{k_1}_{s_1}$ $ \times$ $\cdots$ $\times$ $ \Theta^{k_l}_{s_l}$ $ \times$ $ (S^2_{\mathrm{eq}})^{n - \sum_i k_i}$ $ \subset $ $(\mathbb{C}P^1)^n$, with $s_i \in [1/2,1)$ and $k_i \in 2\mathbb{Z}_{>0}$, $\sum_i k_i \le n$ we get several $l$-dimensional families of non-displaceable Lagrangian tori. We also show that there exists partial symplectic quasi-states $\zeta^{\mathfrak{b}_s}_{\textbf{e}_s}$ and linearly independent homogeneous Calabi quasimorphims $\mu^{\mathfrak{b}_s}_{\textbf{e}_s}$ or which $\Theta^{2m}_s$ are $\zeta^{\mathfrak{b}_s}_{\textbf{e}_s}$-superheavy and $\mu^{\mathfrak{b}_s}_{\textbf{e}_s}$-superheavy. We also prove a similar result for $(\mathbb{C}P^2 3\bar{\mathbb{C}P^2}, \omega_\epsilon)$, where $\{\omega_\epsilon; 0 < \epsilon < 1\}$ is a family of symplectic forms in $\mathbb{C}P^2 3\bar{\mathbb{C}P^2}$, for which $\omega_{1/2}$ is monotone.