ИСТИНА

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 construction and the topology of moment-angle complexes, we shall concentrate on several interesting geometric properties of moment-angle manifolds, emphasising complex-analytic, symplectic and Lagrangian aspects. Different parts of this talk are based on joint works with Victor Buchstaber, Andrei Mironov, Yuri Ustinovsky and Mikhail Verbitsky.