$T^n$-ACTION ON A GRASSMANN MANIFOLD $G_{n,2}$ AND RESOLUTION OF ITS SINGULARITIES

The complex Grassmann manifolds $G_{n,2}$ of complex 2-planes in $\mathbb{C}^n$ can be endowed with the canonical action of the compact torus $T^n$ and the standard moment map $\mu\colon G_{n,2} \to \mathbb{R}^n$, which is defined in terms of Plucker coordinates, which is $T^n$ invariant and its image is the hypersimplex $\Delta_{n,2}$.
In this talk we explain our explicit construction of the model for the orbit space $G_{n,2}/T^n$. This model is the space $U_n = \Delta_{n,2} \times \mathcal{F}_n$ together with the projection $p_n \colon U_n \to \Delta_{n,2}$ such that the orbit space $G_{n,2}/T^n$ is the quotient space of $U_n$ by the relation defined by the map $p_n$. The space $\mathcal{F}_n$ we call an universal space of parameters and it is a smooth manifold which we obtain by the method of algebraic geometry known as the wonderful compactification. The space $U_n$ is a manifold with singular corners, while the space $G_{n,2}/T^n$ is not generally a topological manifold as the case $n = 5$ demonstrates.
The critical points of the moment map $\mu$ as defined in mathematical analysis we proved to coincide with those points in $G_{n,2}$ which have a non-trivial stabilizer in $T^{n-1} = T^n/\mathrm{diag}(T^n)$. Therefore, the notion of a critical point can be extended to the orbit space $G_{n,2}/T^n$. The definition of the universal space of parameters implies that to any Plucker stratum $W_{\sigma} \subset G_{n,2}$ it can be assigned the virtual space of parameters $\tilde{F}_\sigma \subset \mathcal{F}_n$ and the projection $p_{\sigma} \colon \tilde{F}_{\sigma} \to F_{\sigma}$, where $F_{\sigma} = W_{\sigma}/(\mathbb{C}^*)^n$. The space $U_n$ can be considered as a singular fiber space over $G_{n,2}/T^n$, whose fiber over $W_{\sigma}/T^n$ is given by the kernel of the projection $p_{\sigma} \colon \tilde{F}_{\sigma} \to F_{\sigma}$. In this context, we introduce the notion of a singular point in $G_{n,2}/T^n$ as a point of a stratum whose fiber is non trivial. Thus, we show that the space $U_n$ resolves the singularities on $G_{n,2}/T^n$ including the critical points, since we prove that all critical points are singular as well for $n \ge 5$.
The talk is based on the results jointly obtained with Victor M. Buchstaber.
REFERENCES
[1] V. M. Buchstaber and S. Terzić, Topology and geometry of the canonical action of $T^4$ on the complex Grassmannian $G_{4,2}$ and the complex projective space $\mathbb{C}P^5$, Moscow Mathematical Journal 16 (2016), no. 2, 237–273.
[2] V. M. Buchstaber and S. Terzić, Toric topology of the complex Grassmann manifolds, Moscow Mathematical Journal 19 (2019), no. 3, 397–463.
[3] V. M. Bukhshtaber and S. Terzich, The foundations of (2n, k)-manifolds, (Russian) ; translated from Mattem- aticheskii Sbornik 210 (2019), no. 4, 41–86, Sbornik Mathematics 210.
[4] V. M. Buchstaber and S. Terzić, A resolution of singularities for the orbit spaces $G_{n,2}/T^n$, to appear in Proceed- ings of the Steklov Institute of Mathematics
[5] V. M. Buchstaber and S. Terzić, The orbit spaces $G_{n,2}/T^n$ and the Chow quotients $G_{n,2}//(\mathbb{C}^*)^n$ of the Grassmann manifolds $G_{n,2}$, arXiv:2104.08858 (2021).