On the Structure of Certain Natural Cones over Moduli Spaces of Genus-One Holomorphic Maps

TL;DR

This paper demonstrates that certain cones over moduli spaces of genus-one holomorphic maps into projective space have well-defined Euler classes, extending known results from genus-zero cases to genus-one.

Contribution

It establishes the existence of Euler classes for these cones in genus-one, generalizing from the vector bundle case in genus-zero and linking to Gromov-Witten invariants.

Findings

Cones over genus-one moduli spaces have well-defined Euler classes.

The genus-zero case corresponds to vector bundles with known Euler classes.

Genus-one Gromov-Witten invariants relate to these Euler classes.

Abstract

We show that certain naturally arising cones over the main component of a moduli space of $J_0$-holomorphic maps into $P^n$ have a well-defined euler class. We also prove that this is the case if the standard complex structure $J_0$ on $P^n$ is replaced by a nearby almost complex structure $J$. The genus-zero analogue of the cone considered in this paper is always a vector bundle. The genus-zero Gromov-Witten invariant of a projective hypersurface is 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.