A resolution of singularities for the orbit spaces $G_{n,2}/T^n$

TL;DR

This paper introduces a new approach to resolving singularities in the orbit space of the torus action on complex Grassmannians, providing explicit constructions and insights into the combinatorial structure of these spaces.

Contribution

It constructs a smooth manifold with corners that resolves all singular points of the orbit space, offering a new method for understanding torus actions of positive complexity.

Findings

Constructed a smooth manifold with corners $U_n$ resolving singularities

Explicitly described the projection $p_n : U_n o X_n$

Demonstrated a method for general orbit space descriptions for positive complexity actions

Abstract

The problem of the description of the orbit space $X_{n} = G_{n,2}/T^n$ for the standard action of the torus $T^n$ on a complex Grassmann manifold $G_{n,2}$ is widely known and it appears in diversity of mathematical questions. A point $x\in X_{n}$ is said to be a critical point if the stabilizer of its corresponding orbit is nontrivial. In this paper, the notion of singular points of $X_n$ is introduced which opened the new approach to this problem. It is showed that for $n>4$ the set of critical points $\text{Crit}X_n$ belongs to our set of singular points $\text{Sing}X_{n}$, while the case $n=4$ is somewhat special for which $\text{Sing}X_4\subset \text{Crit}X_4$, but there are critical points which are not singular. The central result of this paper is the construction of the smooth manifold $U_n$ with corners, $\dim U_n = \dim X_n$ and an explicit description of the projection $p_{n} : U_{n}\to X_{n}$ which in the defined sense resolve all singular points of the space $X_n$. Thus, we obtain the description of the orbit space $G_{n,2}/T^n$ combinatorial structure. Moreover, the $T^n$-action on $G_{n,2}$ is a seminal example of complexity $(n-3)$ - action. Our results demonstrate the method for general description of orbit spaces for torus actions of positive complexity.