Complex geometry of moment-angle manifolds

TL;DR

This paper explores the complex geometric structures of moment-angle manifolds, constructing transversely Kähler metrics and analyzing subvarieties, revealing their foliation structure and algebraic properties.

Contribution

It introduces methods to construct transversely Kähler metrics on moment-angle manifolds and analyzes the structure of their subvarieties, advancing understanding of their complex geometry.

Findings

Construction of transversely Kähler metrics under certain conditions

Any Kähler subvariety lies within a leaf of the foliation

Generic moment-angle manifolds have algebraic dimension zero

Abstract

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 which is equivariant with respect to the (C*)^m-action. 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 any Kaehler submanifold (or, more generally, a Fujiki class C subvariety) in such a moment-angle manifold is contained in a leaf 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.