Complex geometry of moment-angle manifolds

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.

24 pages, LaTeX, minor changes in version 3

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Cited by in corpus (6)

• Geometric structures on moment-angle manifolds
• On the Structure of Hermitian Manifolds with Semipositive Griffiths Curvature
• The complex geometry of a hypothetical complex structure on $S^6$
• Basic cohomology of canonical holomorphic foliations on complex moment-angle manifolds
• Dolbeault cohomology of complex manifolds with torus action
• Exponential actions defined by vector configurations, Gale duality, and moment-angle manifolds