Abstract
We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for \({\mathbb {C}}P^2\# \overline{{\mathbb {C}}P^2}\), \({\mathbb {C}}P^2\# 2\overline{{\mathbb {C}}P^2}\), 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 \({\mathbb {C}}P^2\# \overline{{\mathbb {C}}P^2}\) also has 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 \(\Theta ^{n_1,n_2,n_3}_{p,q,r} \subset X\) 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 \(\partial (X \setminus \Sigma )\).
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The author is supported by the Herschel Smith postdoctoral fellowship from the University of Cambridge.
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