On the structure of certain natural cones over moduli spaces of genus-one holomorphic maps

We show that certain naturally arising cones over the main component of a moduli space of J0-holomorphic maps into Pn have a well-defined Euler class. We also prove that this is the case if the standard complex structure J0 on Pn is replaced by a nearby almost complex structure J. The genus-zero analogue of the cone considered in this paper is a vector bundle. The genus-zero Gromov–Witten invariant of a projective complete intersection can be viewed as the Euler class of such a vector bundle. As shown in a separate paper, this is also the case for the “genus-one part” of the genus-one GW-invariant. The remaining part is a multiple of the genus-zero GW-invariant.