Toric topology of the complex Grassmann manifolds

The family of the complex Grassmann manifolds $G_{n,k}$ with the canonical action of the torus $T^n=\mathbb{T}^{n}$ and the analogue of the moment map $\mu \colon G_{n,k}\to \Delta _{n,k}$ for the hypersimplex $\Delta _{n,k}$, is well known. In this paper we study the structure of the orbit space $G_{n,k}/T^n$ by developing the methods of toric geometry and toric topology. We use a subdivision of $G_{n,k}$ into the strata $W_{\sigma}$. Relying on this subdivision we determine all regular and singular points of the moment map $\mu$, introduce the notion of the admissible polytopes $P_\sigma$ such that $\mu (W_{\sigma}) = \circ{P}_{\sigma}$ and the notion of the spaces of parameters $F_{\sigma}$, which together describe $W_{\sigma}/T^{n}$ as the product $\circ{P}_{\sigma} \times F_{\sigma}$. To find the appropriate topology for the set $\bigcup_{\sigma} \circ{P}_{\sigma} \times F_{\sigma}$ we introduce also the notions of the universal space of parameters $\tilde{\mathcal{F}}$ and the virtual spaces of parameters $\tilde{F}_{\sigma}\subset \tilde{\mathcal{F}}$ such that there exist the projections $\tilde{F}_{\sigma}\to F_{\sigma}$. Having this in mind, we propose a method for the description of the orbit space $G_{n,k}/T^n$. The existence of the action of the symmetric group $S_{n}$ on $G_{n,k}$ simplifies the application of this method. In our previous paper we proved that the orbit space $G_{4,2}/T^4$, which is defined by the canonical $T^4$-action of complexity $1$, is homeomorphic to $\partial \Delta _{4,2}\ast \mathbb{C} P^1$. We prove in this paper that the orbit space $G_{5,2}/T^5$, which is defined by the canonical $T^5$-action of complexity $2$, is homotopy equivalent to the space which is obtained by attaching the disc $D^8$ to the space $\Sigma ^{4}\mathbb{R} P^2$ by the generator of the group $\pi _{7}(\Sigma ^{4}\mathbb{R} P^2)=\mathbb{Z}_{4}$. In particular, $(G_{5,2}/G_{4,2})/T^5$ is homotopy equivalent to $\partial \Delta _{5,2}\ast \mathbb{C} P^2$. The methods and the results of this paper are very important for the construction of the theory of $(2l,q)$-manifolds we have been recently developing, and which is concerned with manifolds $M^{2l}$ with an effective action of the torus $T^{q}$, $q\leq l$, and an analogue of the moment map $\mu \colon M^{2l}\to P^{q}$, where $P^{q}$ is a $q$-dimensional convex polytope.