An obstruction bundle relating Gromov-Witten invariants of curves and Kahler surfaces

TL;DR

This paper links Gromov-Witten invariants of Kahler surfaces to local invariants of curves via an obstruction bundle, simplifying complex surface calculations to curve-based computations.

Contribution

It introduces an obstruction bundle framework that relates surface GW invariants to curve invariants, extending previous results on local GW invariants.

Findings

Gromov-Witten invariants of surfaces can be expressed through local invariants of curves.

Obstruction bundle approach reduces complex surface calculations to curve GW theory.

Provides a geometric interpretation of local GW invariants in terms of stable maps into curves.

Abstract

In [LP] the authors defined symplectic "Local Gromov-Witten invariants" associated to spin curves and showed that the GW invariants of a Kahler surface X with p_g>0 are a sum of such local GW invariants. This paper describes how the local GW invariants arise from an obstruction bundle (in the sense of Taubes) over the space of stable maps into curves. Together with the results of [LP], this reduces the calculation of the GW invariants of complex surfaces to computations in the GW theory of curves.