Geodesic
Pseudotoric structures: Lagrangian~submanifolds and Lagrangian fibrations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 72 (2017) no. 3, pp. 513-546

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This survey presents a generalization of the notion of a toric structure on a compact symplectic manifold: the notion of a pseudotoric structure. The language of these new structures appears to be a convenient and natural tool for describing many non-standard Lagrangian submanifolds and cycles (Chekanov's exotic tori, Mironov's cycles in certain particular cases, and others) as well as for constructing Lagrangian fibrations (for example, special fibrations in the sense of Auroux on Fano varieties). Known properties of pseudotoric structures and constructions based on these properties are discussed, as well as open problems whose solution may be of importance in symplectic geometry and mathematical physics. Bibliography: 28 titles.
Keywords: symplectic manifold, Lagrangian submanifold, Lagrangian fibration, toric manifold, exotic Lagrangian tori.
Mots-clés : Delzant polytope 
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     author = {N. A. Tyurin},
     title = {Pseudotoric structures: {Lagrangian~submanifolds} and {Lagrangian} fibrations},
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N. A. Tyurin. Pseudotoric structures: Lagrangian~submanifolds and Lagrangian fibrations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 72 (2017) no. 3, pp. 513-546. http://geodesic.mathdoc.fr/item/RM_2017_72_3_a3/