Infinitely many monotone Lagrangian tori in Del Pezzo surfaces

Infinitely many monotone Lagrangian tori in Del Pezzo surfaces

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Renato Vianna, Cambridge University

Joint Seminar with CU at CU.    We will describe how to get almost toric fibrations for all del Pezzo surfaces (endowed with monotone symplectic form), in particular for $\mathbb{CP}^2\#k\overline{\mathbb{CP}}^2$ for $4\le k \le 8$, where there is no toric fibrations. From there, we will be able to construct infinitely many monotone Lagrangian tori. We are able to prove that these tori give rise to infinitely many symplectomorphism classes in $\mathbb{CP}^2\#k\overline{\mathbb{CP}}^2$ for $0 \le k \le 8$, $k \neq 2$, and in $\mathbb{CP}^1 \times \mathbb{CP}^1$. Using the recent work of Pascaleff-Tonkonog one can conclude the same for $\mathbb{CP}^2\#2\overline{\mathbb{CP}}^2$.   Some Markov like equations appear. These equations also appear in the work of Haking-Porokhorov regarding degeneration of surfaces to weighted projective spaces and on the work of Karpov-Nogin regarding 3-block collection of exceptional sheaves in del Pezzo surfaces.