Abstract:
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.
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