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Uspekhi Mat. Nauk, 2013, Volume 68, Issue 3(411), Pages 111–186 (Mi umn9518)  

This article is cited in 9 scientific papers (total in 9 papers)

Geometric structures on moment-angle manifolds

T. E. Panovabcd

a Yaroslavl' State University
b Institute for Theoretical and Experimental Physics
c Institute for Information Transmission Problems of the Russian Academy of Sciences
d Moscow State University

Abstract: A moment-angle complex $\mathscr{Z}_{\mathscr{K}}$ is a cell complex with a torus action constructed from a finite simplicial complex ${\mathscr{K}}$. When this construction is applied to a triangulated sphere ${\mathscr{K}}$ or, in particular, to the boundary of a simplicial polytope, the result is a manifold. Moment-angle manifolds and complexes are central objects in toric topology, and currently are gaining much interest in homotopy theory and complex and symplectic geometry. The geometric aspects of the theory of moment-angle complexes are the main theme of this survey. Constructions of non-Kähler complex-analytic structures on moment-angle manifolds corresponding to polytopes and complete simplicial fans are reviewed, and invariants of these structures such as the Hodge numbers and Dolbeault cohomology rings are described. Symplectic and Lagrangian aspects of the theory are also of considerable interest. Moment-angle manifolds appear as level sets for quadratic Hamiltonians of torus actions, and can be used to construct new families of Hamiltonian-minimal Lagrangian submanifolds in a complex space, complex projective space, or toric varieties.
Bibliography: 59 titles.

Keywords: moment-angle manifold, Hermitian quadrics, simplicial fans, simple polytopes, non-Kähler complex manifolds, Hamiltonian-minimal Lagrangian submanifolds.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation МД-111.2013.1
Russian Foundation for Basic Research 12-01-00873
Dynasty Foundation


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English version:
Russian Mathematical Surveys, 2013, 68:3, 503–568

Bibliographic databases:

MSC: Primary 57R19, 57R17; Secondary 14M25, 32Q55, 52B35, 53D12
Received: 06.02.2013

Citation: T. E. Panov, “Geometric structures on moment-angle manifolds”, Uspekhi Mat. Nauk, 68:3(411) (2013), 111–186; Russian Math. Surveys, 68:3 (2013), 503–568

Citation in format AMSBIB
\by T.~E.~Panov
\paper Geometric structures on moment-angle manifolds
\jour Uspekhi Mat. Nauk
\yr 2013
\vol 68
\issue 3(411)
\pages 111--186
\jour Russian Math. Surveys
\yr 2013
\vol 68
\issue 3
\pages 503--568

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    This publication is cited in the following articles:
    1. Li Cai, “Norm minima in certain Siegel leaves”, Algebr. Geom. Topol., 15:1 (2015), 445–466  crossref  mathscinet  zmath  isi  scopus
    2. Konowski A., Ohno M., Yamaguchi K., “Spaces of algebraic maps from real projective spaces to toric varieties”, J. Math. Soc. Jpn., 68:2 (2016), 745–771  crossref  mathscinet  isi  scopus
    3. Panov T., Ustinovskiy Yu., Verbitsky M., “Complex geometry of moment-angle manifolds”, Math. Z., 284:1-2 (2016), 309–333  crossref  mathscinet  zmath  isi  scopus
    4. V. M. Buchstaber, N. Yu. Erokhovets, M. Masuda, T. E. Panov, S. Park, “Cohomological rigidity of manifolds defined by 3-dimensional polytopes”, Russian Math. Surveys, 72:2 (2017), 199–256  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    5. A. Kozlowski, K. Yamaguchi, “The homotopy type of spaces of rational curves on a toric variety”, Topology Appl., 249 (2018), 19–42  crossref  mathscinet  zmath  isi  scopus
    6. Semyon A. Abramyan, Taras E. Panov, “Higher Whitehead Products in Moment–Angle Complexes and Substitution of Simplicial Complexes”, Proc. Steklov Inst. Math., 305 (2019), 1–21  mathnet  crossref  crossref  mathscinet  isi  elib
    7. Ratiu T., Nguyen Tien Zung, “Presymplectic Convexity and (Ir)Rational Polytopes”, J. Symplectic Geom., 17:5 (2019), 1479–1511  crossref  isi
    8. Kim J.H., “the Torsion of Real Toric Manifolds”, Proc. Amer. Math. Soc., 148:2 (2020), 901–911  crossref  isi
    9. T. Yu. Neretina, “Deistvie svobodnykh kommutiruyuschikh involyutsii na zamknutykh dvumernykh mnogoobraziyakh”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2021, no. 4, 44–47  mathnet
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