Non-Kähler complex structures on moment-angle manifolds and other toric spaces

Moment-angle complexes are spaces acted on by a torus and parameterised by finite simplicial complexes. They are central objects in toric topology, and currently are gaining much interest in homotopy theory. Due the their combinatorial origins, moment-angle complexes also find applications in combinatorial geometry and commutative algebra.
After an introductory part describing the general properties of moment-angle complexes we shall concentrate on the complex-analytic aspects of the theory.
We show that the moment-angle manifolds corresponding to complete simplicial fans admit non-Kähler complex-analytic structures. This generalises the known construction of complexanalytic structures on polytopal moment-angle manifolds, coming from identifying them as LVM-manifolds (or non-degenerate intersections of Hermitian quadrics). The classical series of Hopf and Calabi–Eckmann manifolds are particular examples. We proceed by describing the Dolbeault cohomology and certain Hodge numbers of moment-angle manifolds by applying the Borel spectral sequence to holomorphic principal bundles over toric varieties.
(Joint work with Yuri Ustinovsky.)