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Working paper

Complex geometry of moment-angle manifolds

Panov T., Ustinovsky Y., Verbitsky M.
Moment-angle manifolds provide a wide class of examples of non-Kaehler compact complex manifolds. A complex moment-angle manifold Z is constructed via certain combinatorial data, called a complete simplicial fan. In the case of rational fans, the manifold Z is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori. In general, a complex moment-angle manifold Z is equipped with a canonical holomorphic foliation F and a C*-torus action transitive in the transverse direction. Examples of moment-angle manifolds include Hopf manifolds of Vaisman type, Calabi-Eckmann manifolds, and their deformations. We construct transversely Kaehler metrics on moment-angle manifolds, under some restriction on the combinatorial data. We prove that all Kaehler submanifolds (or, more generally, Fujiki class C subvarieties) in such a moment-angle manifold lie in a compact complex torus contained in a fibre of the foliation F. For a generic moment-angle manifold Z in its combinatorial class, we prove that all subvarieties are moment-angle manifolds of smaller dimension. This implies, in particular, that the algebraic dimension of Z is zero.