{
  "id": "math/0408147",
  "version": "v1",
  "published": "2004-08-11T10:50:44.000Z",
  "updated": "2004-08-11T10:50:44.000Z",
  "title": "A degeneration formula of Gromov-Witten invariants with respect to a curve class for degenerations from blow-ups",
  "authors": [
    "Chien-Hao Liu",
    "Shing-Tung Yau"
  ],
  "comment": "13 pages, 2 figures",
  "categories": [
    "math.AG",
    "hep-th"
  ],
  "abstract": "In two very detailed, technical, and fundamental works, Jun Li constructed a theory of Gromov-Witten invariants for a singular scheme of the gluing form $Y_1\\cup_D Y_2$ that arises from a degeneration $W/{\\Bbb A}^1$ and a theory of relative Gromov-Witten invariants for a codimension-1 relative pair $(Y,D)$. As a summit, he derived a degeneration formula that relates a finite summation of the usual Gromov-Witten invariants of a general smooth fiber $W_t$ of $W/{\\Bbb A}^1$ to the Gromov-Witten invariants of the singular fiber $W_0=Y_1\\cup_D Y_2$ via gluing the relative pairs $(Y_1,D)$ and $(Y_2,D)$. The finite sum mentioned above depends on a relative ample line bundle $H$ on $W/{\\Bbb A}^1$. His theory has already applications to string theory and mathematics alike. For other new applications of Jun Li's theory, one needs a refined degeneration formula that depends on a curve class $\\beta$ in $A_{\\ast}(W_t)$ or $H_2(W_t;{\\Bbb Z})$, rather than on the line bundle $H$. Some monodromy effect has to be taken care of to deal with this. For the simple but useful case of a degeneration $W/{\\Bbb A}^1$ that arises from blowing up a trivial family $X\\times{\\Bbb A}^1$, we explain how the details of Jun Li's work can be employed to reach such a desired degeneration formula. The related set $\\Omega_{(g,k;\\beta)}$ of admissible triples adapted to $(g,k;\\beta)$ that appears in the formula can be obtained via an analysis on the intersection numbers of relevant cycles and a study of Mori cones that appear in the problem. This set is intrinsically determined by $(g,k;\\beta)$ and the normal bundle ${\\cal N}_{Z/X}$ of the smooth subscheme $Z$ in $X$ to be blown up.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-08-11T10:50:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14N35",
      "81T30"
    ],
    "keywords": [
      "degeneration formula",
      "curve class",
      "relative pair",
      "usual gromov-witten invariants",
      "jun lis work"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 656453,
      "adsabs": "2004math......8147L"
    }
  }
}