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

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 ∈ 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 CritX n belongs to our set of singular points SingX n , while the case n = 4 is somewhat special for which SingX 4 ⊂ CritX 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 → 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. 1

The orbit spaces $G_{n,2}/T^n$ and the Chow quotients $G_{n,2}\!/\!/(\mathbb{C} ^{\ast})^{n}$ of the Grassmann manifolds $G_{n,2}$

“…For example M 0,3 is a points, while M 0,4 = CP 1 \ {0, 1, ∞}. Note that the moduli space M 0,n coincides with our space of parameters of the main stratum F n , compare to (5).…”

“…We point out that the matroid polytopes defined in [18] in the case G n,2 coincide with our admissible polytopes [5]. Note that, in our terminology, the Chow quo-tient gives actually a compactification of the space of parameters F n of the main stratum.…”

“…The structures of G n,2 //(C * ) n and G n,2 /T n In [5] we described the orbit space G n,2 /T n in terms of CW complex of admissible polytopes and the universal space of parameters F n . In this description, fundamental role has the chamber decomposition of ∆ n,2 induced by the admissible polytopes.…”

“…Recall [16], [5] that the admissible polytopes define the chamber decomposition for ∆ n,2 : for some subset ω of all admissible sets, C ω is a chamber if…”

“…More explicitly, it turns out that the wonderful compactification of arrangements of subvarieties from [22] can be successfully applied for the compactification of the space of parameters F n of the main stratum W n ⊂ G n,2 . In this way, we obtain the smooth manifold F n which enables us to construct the model U n = F n × ∆ n,2 for the orbit space G n,2 /T n meaning that there exists a continuous surjection p n : U n → G n,2 /T n , see [5]. This Chow quotient is the compactification of the orbit space W n /(C * ) n for the main stratum W n ⊂ G n,2 given in terms of the Chow variety for G n,2 , see [18], [14].…”