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Geometric structures on moment-angle manifolds. (English. Russian original) Zbl 1283.57028
Russ. Math. Surv. 68, No. 3, 503-568 (2013); translation from Usp. Mat. Nauk 68, No. 3, 111-186 (2013).
In this survey several situations are presented in which moment-angle complexes play an important role. The emphasis is put on situations in which a moment-angle complex can be given some geometric structure, i.e., at least a differentiable structure. They include symplectic geometry, in which moment-angle manifolds are important in the classification of toric symplectic manifolds, as well as the algebro-geometric generalization to toric varieties.
Moment-angle manifolds are also relevant in complex geometry; the paper reviews several constructions of a structure of complex manifold on a moment-angle manifold in the even-dimensional case, respectively the product of a moment-angle manifold with a circle in the odd-dimensional case. Then certain holomorphic fibrations with complex moment-angle manifolds as total space with a complex torus as fibre are used to obtain information on the Dolbeault cohomology of complex moment-angle manifolds.
Finally, in the last section, the theory of moment-angle manifolds is used to describe a construction of certain \(H\)-minimal Lagrangian submanifolds in \({\mathbb C}^m\), and to investigate their topological properties.

MSC:
57R19 Algebraic topology on manifolds and differential topology
57R17 Symplectic and contact topology in high or arbitrary dimension
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
32Q55 Topological aspects of complex manifolds
52B35 Gale and other diagrams
53D12 Lagrangian submanifolds; Maslov index
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