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In a previous paper the authors defined symplectic ``Local Gromov-Witten invariants'' associated to spin curves and showed that the GW invariants of a K\"ahler 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 our earlier paper, this reduces the calculation of the GW invariants of elliptic and general-type complex surfaces to computations in the GW theory of curves with additional classes: the Euler classes of the (real) obstruction bundles.

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