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

Victor M. Buchstaber, Svjetlana Terzić

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 $Δ_{n,2}$. The chamber decomposition of $Δ_{n,2}$ 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 $Δ_{n,2}$ 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.

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

Abstract

A symplectic - reduction on a complex Grassmann manifold for the canonical action of the maximal compact torus depends on the - orbit of a maximal chamber in a hypersimplex . The chamber decomposition of 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 - reductions on for the standard moment map in terms of the chamber decomposition of and the well known results of Kirwan and Goldin. We recently introduced the Hassett category whose objects are special compactifications of the space . The initial object in this category is the Deligne-Mumford compactification . In this paper we describe the cohomology rings of various compactifications for , which correspond to the objects of the Hassett category. Our results show that even in the case there are objects in the Hassett category with different cohomology rings.
Paper Structure (15 sections, 29 theorems, 106 equations)

This paper contains 15 sections, 29 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Symplectic forms, invariant complex structures, moment maps
  4. Cohomology of symplectic reductions for complete complex flag manifolds and complex Grassmann manifolds
  5. Combinatorial description of the kernel of the Kirwan map.
  6. Cohomology of a symplectic reduction on $G_{n,2}$ and the chamber decomposition of $\Delta _{n,2}$
  7. Cohomology of spaces from the Hassett category
  8. Computation for $n=4$
  9. Cohomology of spaces from Hassett category $\stackrel{\circ}{\mathcal{H}}_{0, 5}$
  10. Comparison with Chow ring and toric techniques
  11. Comparison with the Chow ring of $\mathcal{M}_{0, 5}$
  12. Comparison with the Chow ring $\mathcal{M}_{0, \mathcal{A}}$
  13. Comparisons with a toric manifold
  14. The case of $\bar{L}_{0, 5}$
  15. The case $\overline{\mathcal{M}}_{0, 5}$

Key Result

Theorem 1

The cohomology of $\mathcal{O}_{\lambda}//T(\xi)$ is isomorphic to the ring for all $\nu$ and $\tau$ such that $\sum _{i=k+1}^{n}\lambda _{\nu (i)} < \sum _{i=k+1}^{n}\xi _{\tau (i)}$ for some $k=1, \ldots , n-1$ and $\deg x_i = \deg u_i =2$.

Theorems & Definitions (49)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • Theorem 2
  • Remark 5
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • ...and 39 more