{
  "id": "2311.15998",
  "version": "v1",
  "published": "2023-11-27T16:46:02.000Z",
  "updated": "2023-11-27T16:46:02.000Z",
  "title": "Beglund-Hübsch transpose and Einstein metrics on rational homology 7-spheres",
  "authors": [
    "Jaime Cuadros Valle",
    "Ralph R. Gomez",
    "Joe Lope Vicente"
  ],
  "comment": "24 pages, 1 table",
  "categories": [
    "math.DG"
  ],
  "abstract": "We show that links of invertible polynomials coming from the Johnson and Koll\\'ar list of K\\\"ahler-Einstein 3-folds that are rational homology 7-spheres remain rational homology 7-spheres under the so-called Berglund-H\\\"ubsch (BH) transpose rule coming from classical mirror symmetry constructions. Actually, the BH-transpose rule produces twins, that is, links with same degree, Milnor number and homology H_3, with the exception of iterated Thom-Sebastiani sums of singularities of chain and cycle type, where the torsion and the Milnor number may vary. This BH-transpose rule not only gives a framework to better understand the existence of SasakiEinstein twins but also gives a mechanism for producing new examples of Sasaki-Einstein twins in the rational homology 7 -sphere setting. We also give reasonable conditions for a Sasaki-Einstein rational homology 7-sphere to remain Sasaki-Einstein under the BH-transpose rule. In particular, we found 75 new examples of Sasaki-Einstein rational homology 7-spheres arising as links of not well-formed hypersurface singularities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-27T16:46:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C25",
      "57R60"
    ],
    "keywords": [
      "einstein metrics",
      "beglund-hübsch transpose",
      "sasaki-einstein rational homology",
      "bh-transpose rule produces twins",
      "milnor number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}