On the Genus-One Gromov-Witten Invariants of Complete Intersections

We state and prove a long-elusive relation between genus-one Gromov-Witten of a complete intersection and twisted Gromov-Witten invariants of the ambient projective space. As shown in a previous paper, certain naturally arising cones of holomorphic vector bundle sections over the main component M 0 1,k(P , d) of the moduli space of stable genus-one holomorphic maps into P have a well-defined euler class. In this paper, we extend this result to moduli spaces of perturbed, in a restricted way, J-holomorphic maps. This extension is used to show that these cones are the correct genus-one analogues of the vector bundles relating genus-zero Gromov-Witten invariants of a complete intersection to those of the ambient projective space. A relationship for highergenus invariants is conjectured as well.