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 DS 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 DS satisfies a duality relation identical to that satisfied by the Gromov invariant Gr introduced by Clifford Taubes, which led Smith to conjecture that DS=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.