{
  "id": "0709.1710",
  "version": "v3",
  "published": "2007-09-11T20:05:32.000Z",
  "updated": "2008-09-11T16:15:28.000Z",
  "title": "Symmetries and exotic smooth structures on a $K3$ surface",
  "authors": [
    "Weimin Chen",
    "Slawomir Kwasik"
  ],
  "comment": "46 pages, final version, Journal of Topology, to appear",
  "categories": [
    "math.GT",
    "math.SG"
  ],
  "abstract": "Smooth and symplectic symmetries of an infinite family of distinct exotic $K3$ surfaces are studied, and comparison with the corresponding symmetries of the standard $K3$ is made. The action on the $K3$ lattice induced by a smooth finite group action is shown to be strongly restricted, and as a result, nonsmoothability of actions induced by a holomorphic automorphism of a prime order $\\geq 7$ is proved and nonexistence of smooth actions by several $K3$ groups is established (included among which is the binary tetrahedral group $T_{24}$ which has the smallest order). Concerning symplectic symmetries, the fixed-point set structure of a symplectic cyclic action of a prime order $\\geq 5$ is explicitly determined, provided that the action is homologically nontrivial.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2008-09-11T16:15:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57S15",
      "57R55",
      "57R17"
    ],
    "keywords": [
      "exotic smooth structures",
      "smooth finite group action",
      "symplectic symmetries",
      "prime order",
      "symplectic cyclic action"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0709.1710C"
    }
  }
}