Complex geometry of manifolds with torus action

Moment-angle manifolds provide a wide class of examples of non-Kaehler compact complex manifolds with a holomorphic torus action. A complex moment-angle manifold $\mathcal{Z}$ is constructed via a 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 this case, the invariants of the complex structure of $\mathcal{Z}$, such Dolbeault cohomology and the Hodge numbers, can be analysed using the Borel spectral sequence of the holomorphic bundle.
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 algebraic torus action. Examples of moment-angle manifolds include the Hopf manifolds, Calabi-Eckmann manifolds, and their deformations. The holomorphic foliated manifold $(\mathcal{Z},\mathcal{F})$ has been also studied as a model for irrational (“non-commutative”) toric varieties in the works several authors (arXiv:1308.2774, arXiv:2002.03876).
We construct transversely Kaehler metrics on moment-angle manifolds $\mathcal{Z}$, under some restriction on the combinatorial data. We prove that all Kaehler submanifolds in such a moment-angle manifold lie in a compact complex torus contained in a fibre of the foliation $\mathcal{F}$. For a generic moment-angle manifold $\mathcal{Z}$ in its combinatorial class, we prove that all its subvarieties are moment-angle manifolds of smaller dimension. This implies, in particular, that $\mathcal{Z}$ does not have non-constant meromorphic functions, i.e. its algebraic dimension is zero.
Battaglia and Zaffran (arXiv:1108.1637) computed the basic Betti numbers for the canonical holomorphic foliation on a moment-angle manifold $\mathcal{Z}$ corresponding to a shellable fan. They conjectured that the basic cohomology ring in the case of any complete simplicial fan has a description similar to the cohomology ring of a complete simplicial toric variety due to Danilov and Jurkiewicz. We prove the conjecture. The proof uses an Eilenberg-Moore spectral sequence argument; the key ingredient is the formality of the Cartan model for the torus action on $\mathcal{Z}$.
The talk is based on joint works with Hiroaki Ishida, Roman Krutowski, Yuri Ustinovsky and Misha Verbitsky.