Complex geometry of moment-angle manifolds

Abstract

Moment-angle manifolds provide a wide class of examples of non-Kähler compact complex manifolds. A complex moment-angle manifold $\mathcal {Z}$ is constructed via certain combinatorial data, called a complete simplicial fan. In the case of rational fans, the manifold $\mathcal {Z}$ is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori. In general, a complex moment-angle manifold $\mathcal {Z}$ is equipped with a canonical holomorphic foliation ${\mathcal {F}}$ which is equivariant with respect to the $({\mathbb {C}}^\times )^m$-action. Examples of moment-angle manifolds include Hopf manifolds of Vaisman type, Calabi–Eckmann manifolds, and their deformations. We construct transversely Kähler metrics on moment-angle manifolds, under some restriction on the combinatorial data. We prove that any Kähler submanifold (or, more generally, a Fujiki class $\mathcal {C}$ subvariety) in such a moment-angle manifold is contained in a leaf of the foliation ${\mathcal {F}}$. For a generic moment-angle manifold $\mathcal {Z}$ in its combinatorial class, we prove that all subvarieties are moment-angle manifolds of smaller dimension and there are only finitely many of them. This implies, in particular, that the algebraic dimension of $\mathcal {Z}$ is zero.