The Gromov invariant and the Donaldson–Smith standard surface count

Abstract

Simon Donaldson and Ivan Smith recently studied symplectic surfaces in symplectic 4–manifolds $X$ by introducing an invariant $\mathcal{D}\mathcal{S}$ associated to any Lefschetz fibration on blowups of $X$ which counts holomorphic sections of a relative Hilbert scheme that is constructed from the fibration. Smith has shown that $\mathcal{D}\mathcal{S}$ satisfies a duality relation identical to that satisfied by the Gromov invariant $Gr$ introduced by Clifford Taubes, which led Smith to conjecture that $\mathcal{D}\mathcal{S}=Gr$ provided that the fibration has high enough degree. This paper proves that conjecture. The crucial technical ingredient is an argument which allows us to work with curves $C$ in the blown-up 4–manifold that are made holomorphic by an almost complex structure which is integrable near $C$ and with respect to which the fibration is a pseudoholomorphic map.