论文标题
轨道空间的奇异性分辨率$ g_ {n，2}/t^n $
A resolution of singularities for the orbit spaces $G_{n,2}/T^n$
论文作者
论文摘要
轨道空间的描述$ x_ {n} = g_ {n，2}/t^n $对于复杂的grassmann歧管$ g_ {n，2} $在圆环$ t^n $的标准操作中是广为人知的，并且在数学问题的多样性中出现。如果X_ {N} $中的点$ X \如果其相应的轨道的稳定器不繁琐，则是关键点。在本文中，引入了$ x_n $的单数点的概念，这为该问题打开了新的方法。 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. 本文的中心结果是构建具有角落的光滑歧管$ u_n $，$ \ dim u_n = \ dim x_n $以及投影$ p_ {n}的明确描述：u_ {n} \ to x__ {n} $，在定义的所有sisse space of Space of Space of Space $ x_n $中的所有sense sense sense s of x_n $ x_n $。因此，我们获得了轨道空间的描述$ g_ {n，2}/t^n $组合结构。此外，$ g_ {n，2} $上的$ t^n $ -action是复杂性$（n-3）$ action的开创性示例。我们的结果证明了对轨道空间的一般描述的方法，以实现积极复杂性的圆环作用。
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.
