{
  "id": "1707.09325",
  "version": "v1",
  "published": "2017-07-28T16:53:49.000Z",
  "updated": "2017-07-28T16:53:49.000Z",
  "title": "A new construction of compact $G_2$-manifolds by gluing families of Eguchi-Hanson spaces",
  "authors": [
    "Dominic Joyce",
    "Spiro Karigiannis"
  ],
  "comment": "82 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "We give a new construction of compact Riemannian 7-manifolds with holonomy $G_2$. Let $M$ be a torsion-free $G_2$-manifold (which can have holonomy a proper subgroup of $G_2$) such that $M$ admits an involution $\\iota$ preserving the $G_2$-structure. Then $M/{\\langle \\iota \\rangle}$ is a $G_2$-orbifold, with singular set $L$ an associative submanifold of $M$, where the singularities are locally of the form $\\mathbb R^3 \\times (\\mathbb R^4 / \\{\\pm 1\\})$. We resolve this orbifold by gluing in a family of Eguchi-Hanson spaces, parametrized by a nonvanishing closed and coclosed $1$-form $\\lambda$ on $L$. Much of the analytic difficulty lies in constructing appropriate closed $G_2$-structures with sufficiently small torsion to be able to apply the general existence theorem of the first author. In particular, the construction involves solving a family of elliptic equations on the noncompact Eguchi-Hanson space, parametrized by the singular set $L$. We also present two generalizations of the main theorem, and we discuss several methods of producing examples from this construction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-28T16:53:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C26",
      "53C29"
    ],
    "keywords": [
      "construction",
      "gluing families",
      "singular set",
      "analytic difficulty lies",
      "noncompact eguchi-hanson space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 82,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}