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Uspekhi Mat. Nauk, 2017, Volume 72, Issue 3(435), Pages 131–169 (Mi umn9764)  

Pseudotoric structures: Lagrangian submanifolds and Lagrangian fibrations

N. A. Tyurinabc

a Joint Institute of Nuclear Research, Bogolyubov Theoretical Physics Laboratory
b National Research University "Higher School of Economics", Laboratory of Algebraic Geometry and Applications
c Moscow State University of Transport

Abstract: 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, Delzant polytope, exotic Lagrangian tori.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 5-100
This paper was written with the support of the programme “Increasing the Competitiveness of Leading Universities of the Russian Federation” (project no. 5-100).


DOI: https://doi.org/10.4213/rm9764

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English version:
Russian Mathematical Surveys, 2017, 72:3, 513–546

Bibliographic databases:

UDC: 516.5
MSC: Primary 53D05, 53D12; Secondary 14M15, 14M25, 53D50
Received: 30.01.2017
Revised: 21.02.2017

Citation: N. A. Tyurin, “Pseudotoric structures: Lagrangian submanifolds and Lagrangian fibrations”, Uspekhi Mat. Nauk, 72:3(435) (2017), 131–169; Russian Math. Surveys, 72:3 (2017), 513–546

Citation in format AMSBIB
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