The complex Grassmann manifolds $G_{n,2}$ is a well known class of Grassmannians widely studied in algebraic geometry and topology. An important problem we study here is the canonical action of the compact torus $T^{n}$ on $G_{n,2}$. This action has complexity $n-3$ since the torus $T^{n-1}$ acts effectively. We use the stratification of $G_{n,2}$ defined by the Plucker coordinates and the existence of the moment map $\mu : G_{n,2}\to \mathbb{R} ^{n}$ whose image is the hypersimplex $\Delta _{n,2}$. Any of the strata $W_{\sigma}$ maps by the moment map $\mu$ to the interior of some subpolytope $P_{\sigma}$ in $\Delta _{n,2}$, which we call an admissible polytope. Among all of these strata a special role plays the main stratum $W$ which is a dense set in $G_{n,2}$ and whose admissible polytope is $\Delta _{n,2}$. Any stratum $W_{\sigma}$ is $T^{n}$-invariant and the orbit space $W_{\sigma}/T^{n}$ is homeomorphic to $\stackrel{\circ}{P}_{\sigma}\times F_{\sigma}$ for some topological space $F_{\sigma}$, which we call the space of parameters for $W_{\sigma}$. In particular, the space of parameters of the main stratum $W$ we denote by $F$. Moreover, the fact that $W/T^n \cong \stackrel{\circ}{\Delta}_{n,2}\times F$ is a dense set in $G_{n,2}/T^{n}$ suggests that there should exist a compactification $\mathcal{F}$ of $F$ and the projection $H : \Delta _{n,2}\times \mathcal{F} \to G_{n,2}/T^n$ such that the space $\Delta _{n,2}\times \mathcal{F}$ quotiented by the equivalence relation defined by $H$ is homeomorphic to $G_{n,2}/T^n$. Such space $\mathcal{F}$ we call the universal space of parameters for $G_{n,2}$. From the same reason one can assign to any stratum $W_{\sigma}$ a subspace $\tilde{F}_{\sigma}$ in $\mathcal{F}$, which we call the virtual space of parameters of the stratum $W_{\sigma}$. In this way $\mathcal{F} = \cup _{\sigma}\tilde{F}_{\sigma}$.

These new notions such as admissible polytopes, spaces of parameters, virtual spaces of parameters and universal space of parameters are introduced and studied in [1], [2].

In the talk we propose a new approach for the description of the admissible polytopes for $T^n$-action on $G_{n,2}$ in terms of the hyperplane arrangements in $\mathbb{R} ^{n-1} = \{(x_1,\ldots ,x_n)\in \mathbb{R} ^{n} | x_1+\ldots +x_n=2\}$ given by $x_{i_1}+x_{i_2}=1, \ldots ,x_{i_1}+\ldots +x_{i_l}=1$, where $1\leq i_1<\ldots <i_l\leq [\frac{n}{2}]$. Moreover, we discuss the universal space of parameters and the virtual spaces of parameters in terms of such description. The cases $n=4,5,6$ will be analyzed in detail.

This is joint work with Victor M. Buchstaber

References

[1] V. M. Buchstaber and S. Terzic, The foundations of $(2n;k)$-manifolds, Mat. Sbornik, Vol. 210, no.4, (2019), 508--549.

[2] V. M. Buchstaber and S. Terzic, Toric Topology of the Complex Grassmann Manifolds, Moscow Math. Jour. Vol.9, Issue 3, (2019), 397--463.