Geometric structures on moment-angle manifolds (Q2854102)

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.NEWLINENEWLINEMoment-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.NEWLINENEWLINEFinally, 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.