Infinitely many monotone Lagrangian tori in del Pezzo surfaces

الأصل · EN

We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for CP² # 1 CP² and CP² # 2 CP², 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 CP² # k CP², for 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 CP² # 1 CP² also have infinitely many monotone Lagrangian tori up to symplectomorphism and we conjecture that the same holds for CP² # 2 CP². Finally, the Lagrangian tori Θⁿ¹,ⁿ²,ⁿ³ₚ,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 Σ. We argue that Θⁿ¹,ⁿ²,ⁿ³ₚ,q,ᵣ give rise to infinitely many exact Lagrangian tori in X Σ, even after attaching the positive end of a symplectization to the boundary of X Σ.