{
  "id": "2002.09231",
  "version": "v1",
  "published": "2020-02-21T11:20:21.000Z",
  "updated": "2020-02-21T11:20:21.000Z",
  "title": "A construction of $G_2$-manifolds from K3 surfaces with a $\\mathbb{Z}^2_2$-action",
  "authors": [
    "Frank Reidegeld"
  ],
  "comment": "37 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "A product of a K3 surface $S$ and a flat 3-dimensional torus $T^3$ is a manifold with holonomy $SU(2)$. Since $SU(2)$ is a subgroup of $G_2$, $S\\times T^3$ carries a torsion-free $G_2$-structure. We assume that $S$ admits an action of $\\mathbb{Z}^2_2$ with certain properties. There are several possibilities to extend this action to $S\\times T^3$. A recent result of Joyce and Karigiannis allows us to resolve the singularities of $(S\\times T^3)/\\mathbb{Z}^2_2$ such that we obtain smooth $G_2$-manifolds. We classify the quotients $(S\\times T^3)/\\mathbb{Z}^2_2$ under certain restrictions and compute the Betti numbers of the corresponding $G_2$-manifolds. Moreover, we study a class of quotients by a non-abelian group. Several of our examples have new values of $(b^2,b^3)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-21T11:20:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C29",
      "14J28"
    ],
    "keywords": [
      "k3 surface",
      "construction",
      "betti numbers",
      "non-abelian group",
      "torsion-free"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}