{
  "id": "1510.07216",
  "version": "v1",
  "published": "2015-10-25T08:08:31.000Z",
  "updated": "2015-10-25T08:08:31.000Z",
  "title": "Upper bounds for the dimension of tori acting on GKM manifolds",
  "authors": [
    "Shintaro Kuroki"
  ],
  "comment": "22 pages, 9 figures",
  "categories": [
    "math.GT",
    "math.CO"
  ],
  "abstract": "The aim of this paper is to give an upper bound for the dimension of a torus $T$ which acts on a GKM manifold $M$ effectively. In order to do that, we introduce a free abelian group of finite rank, denoted by $\\mathcal{A}(\\Gamma,\\alpha,\\nabla)$, from an (abstract) $(m,n)$-type GKM graph $(\\Gamma,\\alpha,\\nabla)$. Here, an $(m,n)$-type GKM graph is the GKM graph induced from a $2m$-dimensional GKM manifold $M^{2m}$ with an effective $n$-dimensional torus $T^{n}$-action, say $(M^{2m},T^{n})$. Then it is shown that $\\mathcal{A}(\\Gamma,\\alpha,\\nabla)$ has rank $\\ell(> n)$ if and only if there exists an $(m,\\ell)$-type GKM graph $(\\Gamma,\\widetilde{\\alpha},\\nabla)$ which is an extension of $(\\Gamma,\\alpha,\\nabla)$. Using this necessarily and sufficient condition, we prove that the rank of $\\mathcal{A}(\\Gamma,\\alpha,\\nabla)$ for the GKM graph of $(M^{2m},T^{n})$ gives an upper bound for the dimension of a torus which can act on $M^{2m}$ effectively. As an application, we compute the rank of $\\mathcal{A}(\\Gamma,\\alpha,\\nabla)$ of the complex Grassmannian of $2$-planes $G_{2}(\\mathbb{C}^{n+2})$ with some effective $T^{n+1}$-action, and prove that the $T^{n+1}$-action on $G_{2}(\\mathbb{C}^{n+2})$ is the maximal effective torus action.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-25T08:08:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57S25"
    ],
    "keywords": [
      "upper bound",
      "type gkm graph",
      "tori acting",
      "maximal effective torus action",
      "free abelian group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151007216K"
    }
  }
}