{
  "id": "math/0510528",
  "version": "v1",
  "published": "2005-10-25T17:39:28.000Z",
  "updated": "2005-10-25T17:39:28.000Z",
  "title": "Orbifold Cohomology of ADE-singularities",
  "authors": [
    "Fabio Perroni"
  ],
  "comment": "PhD thesis at SISSA, Trieste (Italy), 113 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "We study Ruan's \"cohomological crepant resolution conjecture\" (see math.AG/0108195) for orbifolds with transversal ADE singularities. Let [Y] be such an orbifold, Y its coarse moduli space and Z the crepant resolution of Y. Following Ruan [math.AG/0108195], we have a deformation of the cohomology ring H^*(Z) using some Gromov-Witten invariants of Z. The resulting family of rings will be denoted by H^*(Z)(q_1,...,q_n), where q_1,...,q_n are complex parameters. In the A_n case we compute both the orbifold cohomology ring H^*_{orb}([Y]) and the family H^*(Z)(q_1,...,q_n). The former is achieved in general, the later up to an explicit conjecture on the Gromov-Witten invariants which is verified under additional, technical assumptions. We construct an explicit isomorphism between H^*_{orb}([Y]) and H^*(Z)(-1) in the A_1 case, verifying Ruan's conjecture. In the A_2 case, the family H^*(q_1,q_2) is not defined for q_1=q_2=-1, so the conjecture should be slightly modified. However we give an explicit isomorphism between H^*_{orb}([Y]) and H^*(Z)(q_1,q_2) with q_1=q_2 be a primitive third root of the unity, thus proving a slightly modified version of Ruan's conjecture. It is natural to conjecture that, in the A_n case, H^*_{orb}([Y]) is isomorphic to H^*(Z)(q_1,...,q_n) with q_1=...=q_n be a primitive (n+1)-th root of the unity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-10-25T17:39:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14N35",
      "14E15"
    ],
    "keywords": [
      "orbifold cohomology",
      "ade-singularities",
      "gromov-witten invariants",
      "explicit isomorphism",
      "coarse moduli space"
    ],
    "tags": [
      "dissertation"
    ],
    "publication": {
      "journal": "Ph.D. Thesis",
      "year": 2005,
      "month": "Oct"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 113,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005PhDT.......297P"
    }
  }
}