Abstract
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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References
N. A. Tyurin, “Pseudotoric structures: Lagrangian submanifolds and Lagrangian fibrations,” Russ. Math. Surv. 72 (3), 513–546 (2017) [transl. from Usp. Mat. Nauk 72 (3), 131–169 (2017)].
N. A. Tyurin, “Universal Maslov class of a Bohr-Sommerfeld Lagrangian embedding into a pseudo-Einstein manifold,” Theor. Math. Phys. 150 (2), 278–287 (2007) [transl. from Teor. Mat. Fiz. 150 (2), 325–337 (2007)].
M. Audin, Torus Actions on Symplectic Manifolds (Birkhäuser, Basel, 2004), Prog. Math. 93.
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This work is supported by the Russian Science Foundation under grant 19-11-00164.
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This article was submitted by the author simultaneously in Russian and English
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2019, Vol. 307, pp. 291–305.
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Tyurin, N.A. Monotonic Lagrangian Tori of Standard and Nonstandard Types in Toric and Pseudotoric Fano Varieties. Proc. Steklov Inst. Math. 307, 267–280 (2019). https://doi.org/10.1134/S0081543819060166
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DOI: https://doi.org/10.1134/S0081543819060166