Infinitely many monotone Lagrangian tori in del Pezzo surfaces

Abstract

We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for $\mathbb{C}P^2 \# 1 \overline{\mathbb{C}P^2}$ and $\mathbb{C}P^2 \# 2 \overline{\mathbb{C}P^2}$ , we are able to get almost toric base diagrams (ATBDs) of triangular shape and prove the existence of infinitely many symplectomorphism (in particular Hamiltonian isotopy) classes of monotone Lagrangian tori in $\mathbb{C}P^2 \# k \overline{\mathbb{C}P^2}$, for k=0,3,4,5,6,7,8. We name these tori $\Theta^{n_1,n_2,n_3}_{p,q,r}$. Using the work of Karpov-Nogin, we are able to classify all ATBDs of triangular shape. We are able to prove that $\mathbb{C}P^2 \# 1 \overline{\mathbb{C}P^2}$ also have infinitely many monotone Lagrangian tori up to symplectomorphism and we conjecture that the same holds for $\mathbb{C}P^2 \# 2 \overline{\mathbb{C}P^2}$ . Finally, the Lagrangian tori $\Theta^{n_1,n_2,n_3}_{p,q,r}$ inside a del Pezzo surface $X$ can be seen as monotone fibres of ATFs, such that, over its edge lies a fixed anticanonical symplectic torus $\Sigma$. We argue that $\Theta^{n_1,n_2,n_3}_{p,q,r}$ give rise to infinitely many exact Lagrangian tori in $X \setminus \Sigma$, even after attaching the positive end of a symplectization to the boundary of $X \setminus \Sigma$.