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This article is cited in 8 scientific papers (total in 8 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.
DOI:
https://doi.org/10.4213/rm9518
Full text:
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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
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http://mi.mathnet.ru/eng/umn9518https://doi.org/10.4213/rm9518 http://mi.mathnet.ru/eng/umn/v68/i3/p111
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