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Geometric structures on moment-angle manifolds

The moment-angle complex Z_K is cell complex with a torus action constructed from a finite simplicial complex K. When this construction is applied to a triangulated sphere 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, complex and symplectic geometry. The geometric aspects of the theory of moment-angle complexes are the main theme of this survey. We review constructions of non-Kahler complex-analytic structures on moment-angle manifolds corresponding to polytopes and complete simplicial fans, and describe invariants of these structures, such as the Hodge numbers and Dolbeault cohomology rings. 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.