Journal of Symplectic Geometry

Volume 16 (2018)

Number 3

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

Pages: 857 – 883



Renato Vianna (Dept. Matemática, Instituto de Matemática, Universidade Federal do Rio de Janeiro (UFRJ), Centro de Tecnologia Cidade Universitária, Rio de Janeiro, Brazil)


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 in [7]. We show that for $n = 2m$ and $s \geq 1/2$ these tori are nondisplaceable. Then by considering $\Theta^{k_1}_{s_1} \times \dotsm \times \Theta^{k_l}_{s_l} \times (S^2_{\mathrm{eq}})^{n-\sum_i k_i} \subset (\mathbb{C}P^1)^n$, with $s_i \in [1/2,1)$ and $k_i \in 2 \mathbb{Z}_{\gt 0}, \sum_i k_i \leq n$ we get several $l$-dimensional families of non-displaceable Lagrangian tori. We also show that there exists partial symplectic quasi-states $\zeta^{\mathfrak{b}_s}_{\mathfrak{e}_s}$ and linearly independent homogeneous Calabi quasimorphims $\mu^{\mathfrak{b}_s}_{\mathfrak{e}_s}$ [18] for which $\Theta^{2m}_s$ are $\zeta^{\mathfrak{b}_s}_{\mathfrak{e}_s}$-superheavy and $\mu^{\mathfrak{b}_s}_{\mathfrak{e}_s}$-superheavy. We also prove a similar result for $(\mathbb{C} P^2 \# 3 \overline{\mathbb{C} P^2}, \omega_{\epsilon})$, where $\lbrace \omega_{\epsilon} ; 0 \lt \epsilon \lt 1 \rbrace$ is a family of symplectic forms in $\mathbb{C} P^2 \# 3 \overline{\mathbb{C} P^2}$, for which $\omega_{1/2}$ is monotone.

The author is supported by the Herschel Smith postdoctoral fellowship from the University of Cambridge.

Received 5 March 2016

Accepted 8 December 2017

Published 26 November 2018