Toric topology of complex Grassmann manifolds

The complex Grassmann manifold $G(n,k)$ of all $k$-dimensional complex linear subspaces in the complex vector space $C^n$ plays the fundamental role in algebraic topology, algebraic and complex geometry, and other areas of mathematics. The manifolds $G(n,1)$ and $G(n,n-1)$ can be identified with the complex projective space $CP(n-1)$. The coordinate-wise action of the compact torus $T^n$ on $C^n$ induces its canonical action on the manifolds $G(n,k)$. The orbit space $CP(n-1)/T^n$ can be identified with the $(n-1)$-dimensional simplex. The description of the combinatorial structure and algebraic topology of the orbit space $G(n,k)/T^n$, where $k$ is not $1$ or $(n-1)$, is a well-known topical problem, which is far from being solved. The talk is devoted to the results in this direction which were recently obtained by methods of toric topology jointly with Svjetlana Terzić (University of Montenegro, Podgorica).