Infinitely many monotone Lagrangian tori in del Pezzo surfaces
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Renato Vianna [1]
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[1] University of Cambridge
University of Cambridge
Cambridge District, Reino Unido
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- Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 23, Nº. 3, 2017, págs. 1955-1996
- Idioma: inglés
- DOI: 10.1007/s00029-017-0312-z
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Resumen
We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for CP2#CP2, CP2#2CP2, 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#kCP2 for k = 0, 3, 4, 5, 6, 7, 8. We name these tori n1,n2,n3 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 CP2#CP2 also has infinitely many monotone Lagrangian tori up to symplectomorphism and we conjecture that the same holds for CP2#2CP2. Finally, the Lagrangian tori n1,n2,n3 p,q,r ⊂ X can be seen as monotone fibres of ATFs, such that, over its edge lies a fixed anticanonical symplectic torus . We argue that n1,n2,n3 p,q,r give rise to infinitely many exact Lagrangian tori in X \ , even after attaching the positive end of a symplectization to ∂(X \ ). Math.
