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Russian Mathematical Surveys, 2017, Volume 72, Issue 3, Pages 513–546
DOI: https://doi.org/10.1070/RM9764 (Mi rm9764) 		 
This article is cited in 3 scientific papers (total in 3 papers)

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
English version PDF (667 kB) Citations (3) Russian version article
References:
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DOI: https://doi.org/10.1070/RM9764
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).
Received: 30.01.2017
Revised: 21.02.2017
Bibliographic databases:
Document Type: Article
UDC: 516.5
MSC: Primary 53D05, 53D12; Secondary 14M15, 14M25, 53D50
Language: English
Original paper language: Russian
Citation: N. A. Tyurin, “Pseudotoric structures: Lagrangian submanifolds and Lagrangian fibrations”, Russian Math. Surveys, 72:3 (2017), 513–546
Citation in format AMSBIB
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\by N.~A.~Tyurin
\paper Pseudotoric structures: Lagrangian~submanifolds and Lagrangian fibrations
\jour Russian Math. Surveys
\yr 2017
\vol 72
\issue 3
\pages 513--546
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Linking options:
  • https://www.mathnet.ru/eng/rm9764
  • https://doi.org/10.1070/RM9764
  • https://www.mathnet.ru/eng/rm/v72/i3/p131
    • This publication is cited in the following 3 articles:
    1. N. A. Tyurin, “Lagrangian geometry of algebraic manifolds”, Phys. Part. Nuclei Lett., 19:4 (2022), 337  crossref
    2. N. A. Tyurin, “Mironov Lagrangian cycles in algebraic varieties”, Sb. Math., 212:3 (2021), 389–398  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. Nikolai A. Tyurin, “Monotonic Lagrangian Tori of Standard and Nonstandard Types in Toric and Pseudotoric Fano Varieties”, Proc. Steklov Inst. Math., 307 (2019), 267–280  mathnet  crossref  crossref  mathscinet  isi  elib
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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    Abstract page:3444
    Russian version PDF:124
    English version PDF:35
    References:78
    First page:37
     
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