The Gromov invariant and the Donaldson{Smith standard surface count
نویسنده
چکیده
Simon Donaldson and Ivan Smith recently studied symplectic surfaces in symplectic 4{manifolds X by introducing an invariant DS associated to any Lefschetz bration on blowups of X which counts holomorphic sections of a relative Hilbert scheme that is constructed from the bration. Smith has shown that DS satis es a duality relation identical to that satis ed by the Gromov invariant Gr introduced by Cli ord Taubes, which led Smith to conjecture that DS = Gr provided that the bration 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 bration is a pseudoholomorphic map. AMS Classi cation numbers Primary: 53D45
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