Geometric structures on moment-angle manifolds

Abstract

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.

60 pages

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• Complex geometry of moment-angle manifolds
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Cited by in corpus (10)

• Characteristic Classes of SL(N)-Bundles and Quantum Dynamical Elliptic R-Matrices
• Complex geometry of moment-angle manifolds
• Higher Whitehead products in moment-angle complexes and substitution of simplicial complexes
• Non-negative curvature and torus actions
• Products and connected sums of spheres as monotone Lagrangian submanifolds
• Norm minima in certain Siegel leaves
• On the moment-angle manifold constructed by Fan, Chen, Ma and Wang
• Non-isotopic monotone Lagrangian submanifolds of ${\mathbb{C}}^n$
• The homotopy type of the space of algebraic loops on a toric variety
• Free commuting involutions on closed two-dimensional surfaces