  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.
