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 CP$^{2}$#CP$^{2}$, CP$^{2}$#2CP$^{2}$, 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 CP$^{2}$#$\textit{k}$CP$^{2}$ for $\textit{k}$=0,3,4,5,6,7,8. We name these tori Θn1,n2,n3p,q,r. Using the work of Karpov-Nogin, we are able to classify all ATBDs of triangular shape. We are able to prove that CP$^{2}$#CP$^{2}$also has infinitely many monotone Lagrangian tori up to symplectomorphism and we conjecture that the same holds for CP$^{2}$#2CP$^{2}$. Finally, the Lagrangian tori Θ$\frac{n1,n2,n3}{p,q,r}$ ⊂$\textit{X}$ can be seen as monotone fibres of ATFs, such that, over its edge lies a fixed anticanonical symplectic torus Σ. We argue that Θ$\frac{n1,n2,n3}{p,q,r}$ give rise to infinitely many exact Lagrangian tori in $\textit{X}$∖Σ, even after attaching the positive end of a symplectization to ∂($\textit{X}$∖Σ).