Toric topology of the complex Grassmann manifolds

The family of the complex Grassmann manifolds $G_{n,k}$ with a canonical action of the torus $T^n={\mathbb{T}}^n$ and the analogue of the moment map $\mu :G_{n,k}\to {\mathrm{\Delta }}_{n,k}$ for the hypersimplex ${\mathrm{\Delta }}_{n,k}$, is well known. In this paper we study the structure of the orbit space $G_{n,k}\mathrm{/}{T}^n$ by developing the methods of toric geometry and toric topology. We use a subdivision of $G_{n,k}$ into the strata $W_{\sigma }$ and determine all regular and singular points of the moment map $\mu$, introduce the notion of the admissible polytopes $P_{\sigma }$ such that $\mu (W_{\sigma}) = \stackrel{\circ}{P_{\sigma}}$andthenotionofthespacesofparameters$F_{\sigma}$,whichtogetherdescribe$W_{\sigma}/T^{n}$astheproduct$\stackrel{\circ}{P_{\sigma}} \times F_{\sigma}$. To find the appropriate topology for the set $\cup {\sigma} \stackrel{\circ}{P{\sigma}} \times F_{\sigma}$weintroducethenotionsoftheuniversalspaceofparameters$\tilde{$\mathcal{F}$}$andthevirtualspacesofparameters$\tilde{F}_{\sigma}\subset \tilde{$\mathcal{F}$}$ such that there exist the projections ${\tilde{F}}_{\sigma }\to F_{\sigma }$. Hence, we propose a method for the description of the orbit space $G_{n,k}\mathrm{/}{T}^n$. Earlier we proved that the orbit space $G_{4,2}\mathrm{/}{T}^4$, defined by the canonical $T^4$-action of complexity $1$, is homeomorphic to $\mathrm{\partial }{\mathrm{\Delta }}_{4,2}\ast \mathbb{C}{P}^1$. We prove here that the orbit space $G_{5,2}\mathrm{/}{T}^5$, defined by the canonical $T^5$-action of complexity $2$, is homotopy equivalent to the space obtained by attaching the disc $D^8$ to the space ${\mathrm{\Sigma }}^4\mathbb{R}{P}^2$ by the generator of the group ${\pi }_7({\mathrm{\Sigma }}^4\mathbb{R}{P}^2)={\mathbb{Z}}_4$. In particular, $(G_{5,2}\mathrm{/}{G}_{4,2})\mathrm{/}{T}^5$ is homotopy equivalent to $\partial \Delta _{5,2}\ast $\mathbb{C}$ P^2$. The methods and the results of this paper are fundaments for our theory of $(2l,q)$-manifolds.