Cohomology of symplectic $T^n$ - reductions and compactifications of $\mathcal{M}_{0, n}$

Abstract

A symplectic $T^n$ - reduction on a complex Grassmann manifold $G_{n,2}$ for the canonical action of the maximal compact torus depends on the $S_n$ - orbit of a maximal chamber in a hypersimplex $\Delta _{n,2}$. The chamber decomposition of $\Delta _{n,2}$ is defined by the admissible polytopes, which can be realized as matroids. In our previous work we described this chamber decomposition by means of the hyperplane arrangement. It is important to note that to any chamber it corresponds the compact space, which is a smooth compact manifold for a chamber of maximal dimension. In this paper we obtain explicit description of the cohomology rings of the symplectic $T^n$ - reductions on $G_{n,2}$ for the standard moment map in terms of the chamber decomposition of $\Delta _{n,2}$ and the well known results of Kirwan and Goldin. We recently introduced the Hassett category whose objects are special compactifications of the space $\mathcal{M}_{0, n}$. The initial object in this category is the Deligne-Mumford compactification $\overline{\mathcal{M}}_{0, n}$. In this paper we describe the cohomology rings of various compactifications for $\mathcal{M}_{0, n}$, which correspond to the objects of the Hassett category. Our results show that even in the case $n=5$ there are objects in the Hassett category with different cohomology rings.