Infinitely many monotone Lagrangian tori in del Pezzo surfaces

By Renato Vianna. Selecta Mathematica, 2016

65 citations recorded by the source corpus.

We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for CP2#CP2¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}P^2\# \overline{{\mathbb {C}}P^2}$$\end{document}, CP2#2CP2¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}P^2\# 2\overline{{\mathbb {C}}P^2}$$\end{document}, 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¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}P^2\# k\overline{{\mathbb {C}}P^2}$$\end{document} for k=0,3,4,5,6,7,8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=0,3,4,5,6,7,8$$\end{document}. We name these tori Θp,q,rn1,n2,n3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta ^{n_1,n_2,n_3}_{p,q,r}$$\end{document}. Using the work of Karpov-Nogin, we are able to classify all ATBDs of triangular shape. We are able to prove that CP2#CP2¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}P^2\# \overline{{\mathbb {C}}P^2}$$\end{document} also has infinitely many monotone Lagrangian tori up to symplectomorphism and we conjecture that the same holds for CP2#2CP2¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}P^2\# 2\overline{{\mathbb {C}}P^2}$$\end{document}. Finally, the Lagrangian tori Θp,q,rn1,n2,n3⊂X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta ^{n_1,n_2,n_3}_{p,q,r} \subset X$$\end{document} can be seen as monotone fibres of ATFs, such that, over its edge lies a fixed anticanonical symplectic torus Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document}. We argue that Θp,q,rn1,n2,n3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta ^{n_1,n_2,n_3}_{p,q,r}$$\end{document} give rise to infinitely many exact Lagrangian tori in X\Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X \setminus \Sigma $$\end{document}, even after attaching the positive end of a symplectization to ∂(X\Σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial (X \setminus \Sigma )$$\end{document}.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/127e2d2e415ca56c6e2ceb3ea941a86babc8ff63

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In Geometry and Topology

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In Mathematical Physics

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  2. James Pascaleff
  3. Honghao Gao
  4. Dmitry Tonkonog

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