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GKM graph locally modeled by $T^{n}\times S^{1}$-action on $T^{*}\mathbb{C}^{n}$ and its graph equivariant cohomology

Shintaro Kuroki, Vikraman Uma

Abstract

We introduce a class of labeled graphs (with legs) which contains two classes of GKM graphs of $4n$-dimensional manifolds with $T^{n}\times S^{1}$-actions, i.e., GKM graphs of the toric hyperK${\rm\ddot{a}}$hler manifolds and of the cotangent bundles of toric manifolds. Under some conditions, the graph equivariant cohomology ring of such a labeled graph is computed. We also give a module basis of the graph equivariant cohomology by using a shelling structure of such a labeled graph and study their multiplicative structure.

GKM graph locally modeled by $T^{n}\times S^{1}$-action on $T^{*}\mathbb{C}^{n}$ and its graph equivariant cohomology

Abstract

We introduce a class of labeled graphs (with legs) which contains two classes of GKM graphs of -dimensional manifolds with -actions, i.e., GKM graphs of the toric hyperKhler manifolds and of the cotangent bundles of toric manifolds. Under some conditions, the graph equivariant cohomology ring of such a labeled graph is computed. We also give a module basis of the graph equivariant cohomology by using a shelling structure of such a labeled graph and study their multiplicative structure.

Paper Structure

This paper contains 24 sections, 28 theorems, 235 equations, 16 figures.

Table of Contents

  1. Introduction
  2. GKM graph locally modeled by $T^{n}\times S^{1}$-action on $T^{*}\mathbb{C}^{n}$
  3. Notations
  4. GKM graph with legs and its graph equivariant cohomology
  5. Some notions of $T^{*}\mathbb{C}^{n}$-modeled GKM graphs
  6. Hyperplane
  7. Pre-halfspace and its Thom class
  8. Opposite side of a pre-halfspace
  9. Halfspace and Ring ${\mathbb{Z}}[\mathcal{G}]$
  10. Ring structure of the graph equivariant cohomology of a $T^{*}\mathbb{C}^{n}$-modeled GKM graph
  11. An $x$-forgetful graph $\mathcal{\widetilde{G}}$
  12. $x$-forgetful graph $\mathcal{\widetilde{G}}$ and its graph equivariant cohomology
  13. The ring structure of $H^{*}(\mathcal{\widetilde{G}})$
  14. The localization map and the injectivity of $\Psi'$
  15. The surjectivity of $\Psi'$
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\mathcal{G}$ be a $2n$-valent $T^{*}\mathbb{C}^{n}$-modeled GKM graph and $\mathbf{L}=\{L_{1},\ \cdots,\ L_{m}\}$ be the set of all hyperplanes in $\mathcal{G}$. Assume that $\mathcal{G}$ satisfies the following two assumptions: Then the following ring isomorphism holds:

Figures (16)

  • Figure 1: These are examples of regular graphs with legs and orientations. The left $2$-valent graph has two legs, on the other hand the right $3$-valent graph has no legs. Note that all edges have two orientations and all legs have only one orientation.
  • Figure 2: $T^{*}\mathbb{C}^{2}$-modeled GKM graphs, where $\langle {\rm e_{1}^{*},\ e_{2}^{*}} \rangle\simeq (\mathfrak{t}^{2})^{*}_{{\mathbb{Z}}}$. In the left and the right figures, we assume $\alpha(\epsilon)=-\alpha(\overline{\epsilon})$ and omit some axial functions which are automatically determined by the definition.
  • Figure 3: The left figure shows a hyperplane of the left GKM graph in Figure \ref{['Figure2']}. The right figure is a $2$-valent GKM subgraph of the left GKM graph in Figure \ref{['Figure2']} but it is not a hyperplane.
  • Figure 4:
  • Figure 5:
  • ...and 11 more figures

Theorems & Definitions (69)

  • Theorem 1.1: Theorem \ref{['main-theorem1']}
  • Remark 2.1
  • Definition 2.2: GKM graph with legs
  • Remark 2.3
  • Definition 2.4: GKM graph locally modeled by $T^{n}\times S^{1}$-action on $T^{*}\mathbb{C}^{n}$
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7: graph equivariant cohomology
  • Definition 3.1: hyperplane
  • ...and 59 more