# Geometric structures on moment-angle manifolds

# Geometric structures on moment-angle manifolds

Moment-angle complexes are spaces acted on by a torus and parametrised by finite simplicial complexes. They are central objects in toric topology, and currently are gaining much interest in the homotopy theory. Due the their combinatorial origins, moment-angle complexes also find applications in combinatorial geometry and commutative algebra. Moment-angle complexes corresponding to simplicial subdivision of spheres are topological manifolds, and those corresponding to simplicial polytopes admit smooth realisations as intersection of real quadrics in $C^m$. After an introductory part describing the general properties of moment-angle complexes we shall concentrate on the complex-analytic and Lagrangian aspects of the theory. We show that the moment-angle manifolds corresponding to complete simplicial fans admit nonKaehler complex-analytic structures. This generalises the known construction of complex-analytic structures on polytopal moment-angle manifolds, coming from identifying them as LVM-manifolds. 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. A new wide family of minimal Lagrangian submanifolds N in $C^m$ or $CP^m$ can be constructed from intersections of real quadrics. These submanifolds have the following topological properties: every N embeds in the corresponding moment-angle manifold Z, and every N is the total space of two different fibrations, one over the torus $T^{m-n}$ with fibre a real moment-angle manifold R, and another over a small cover with fibre a torus. These properties are used to produce new examples of Lagrangian submanifolds with quite complicated topology. Different parts of this talk are based on joint works with Victor Buchstaber, Andrei Mironov and Yuri Ustinovsky.