Gromov-Witten invariants of a quintic threefold and a rigidity conjecture

We show that a widely believed conjecture concerning rigidity of genuszero and genus-one holomorphic curves in Calabi-Yau threefolds implies a relation between the genus-one GW-invariants of a quintic threefold in P⁴ and the genus-zero and genus-one GW-invariants of P⁴. This relation is a special case of a general formula for the genus-one GW-invariants of complete intersections obtained in a previous paper. In contrast to the general case, this paper's derivation is more geometric and makes direct use of the rigidity property. Thus, it provides further evidence for the rigidity conjecture in low genera. On the other hand, this paper also suggests a potential way of disapproving the less commonly believed generalization of the rigidity conjecture to arbitrary genus.