Evidence for a conjecture of Pandharipande
| dc.creator | Bryan, Jim | |
| dc.date | 2000-08-03 | |
| dc.date.accessioned | 2026-07-07T04:36:39Z | |
| dc.date.available | 2026-07-07T04:36:39Z | |
| dc.description | 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. | |
| dc.description | For proceedings of 7th Gokova Geometry and Topology conference; uses gokova.cls | |
| dc.identifier | https://arxiv.org/abs/math/0008036 | |
| dc.identifier | http://arxiv.org/abs/math/0008036 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/59678 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Symplectic Geometry | |
| dc.subject | 14N35,53D45 | |
| dc.title | Evidence for a conjecture of Pandharipande | |
| dc.type | text |