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

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.