Geodesic
Monotonic Lagrangian Tori of Standard and Nonstandard Types in Toric and Pseudotoric Fano Varieties
Informatics and Automation, Algebra, number theory, and algebraic geometry, Tome 307 (2019), pp. 291-305.

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In recent papers we constructed examples of nonstandard Lagrangian tori in compact simply connected toric symplectic manifolds. Using a new “pseudotoric” technique, we explained the appearance of nonstandard Lagrangian tori of Chekanov type and proposed a topological obstruction which separates them from the standard ones. In the present paper we construct nonstandard tori satisfying the Bohr–Sommerfeld condition with respect to the anticanonical class. Then we prove that if there exists a standard monotonic Lagrangian torus in a smooth simply connected toric Fano variety equipped with a canonical symplectic form, then there must exist a monotonic Lagrangian torus of Chekanov type. 
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     author = {Nikolai A. Tyurin},
     title = {Monotonic {Lagrangian} {Tori} of {Standard} and {Nonstandard} {Types} in {Toric} and {Pseudotoric} {Fano} {Varieties}},
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     url = {http://geodesic.mathdoc.fr/item/TRSPY_2019_307_a15/}
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Nikolai A. Tyurin. Monotonic Lagrangian Tori of Standard and Nonstandard Types in Toric and Pseudotoric Fano Varieties. Informatics and Automation, Algebra, number theory, and algebraic geometry, Tome 307 (2019), pp. 291-305. http://geodesic.mathdoc.fr/item/TRSPY_2019_307_a15/

[1] N. A. Tyurin, “Pseudotoric structures: Lagrangian submanifolds and Lagrangian fibrations”, Russ. Math. Surv., 72:3 (2017), 513–546 | DOI | DOI | MR | Zbl

[2] N. A. Tyurin, “Universal Maslov class of a Bohr–Sommerfeld Lagrangian embedding into a pseudo-Einstein manifold”, Theor. Math. Phys., 150:2 (2007), 278–287 | DOI | DOI | MR | Zbl

[3] Audin M., Torus actions on symplectic manifolds, Prog. Math., 93, Birkhäuser, Basel, 2004 | MR | Zbl