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 ${\Theta }_{p,q,r}^{n_1,n_2,n_3}$. 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 ${\Theta }_{p,q,r}^{n_1,n_2,n_3}$ 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 ${\Theta }_{p,q,r}^{n_1,n_2,n_3}$ 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 $X\setminus \Sigma$.

28 pages, 20 figures