{
  "id": "math/0008036",
  "version": "v1",
  "published": "2000-08-03T22:41:52.000Z",
  "updated": "2000-08-03T22:41:52.000Z",
  "title": "Evidence for a conjecture of Pandharipande",
  "authors": [
    "Jim Bryan"
  ],
  "comment": "For proceedings of 7th Gokova Geometry and Topology conference; uses gokova.cls",
  "categories": [
    "math.AG",
    "math.SG"
  ],
  "abstract": "In his paper \"Hodge integrals and degenerate contributions\", Pandharipande studied the relationship between the enumerative geometry of certain 3-folds and the Gromov-Witten invariants. In some good cases, enumerative invariants (which are manifestly integers) can be expressed as a rational combination of Gromov-Witten invariants. Pandharipande speculated that the same combination of invariants should yield integers even when they do not have any enumerative significance on the 3-fold. In the case when the 3-fold is the product of a complex surface and an elliptic curve, Pandharipande has computed this combination of invariants on the 3-fold in terms of the Gromov-Witten invariants of the surface. This computation yields surprising conjectural predictions about the genus 0 and genus 1 Gromov-Witten invariants of complex surfaces. The conjecture states that certain rational combinations of the genus 0 and genus 1 Gromov-Witten invariants are always integers. Since the Gromov-Witten invariants for surfaces are often enumerative (as oppose to 3-folds), this conjecture can often also be interpreted as giving certain congruence relations among the various enumerative invariants of a surface. In this note, we state Pandharipande's conjecture and we prove it for an infinite series of classes in the case of the projective plane blown-up at 9 points. In this case, we find generating functions for the numbers appearing in the conjecture in terms of quasi-modular forms. We then prove the integrality of the numbers by proving a certain a congruence property of modular forms that is reminiscent of Ramanujan's mod 5 congruences of the partition function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2000-08-03T22:41:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14N35",
      "53D45"
    ],
    "keywords": [
      "gromov-witten invariants",
      "complex surface",
      "rational combination",
      "computation yields surprising conjectural predictions",
      "congruence"
    ],
    "tags": [
      "conference paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math......8036B"
    }
  }
}