The Gromov invariant and the Donaldson-Smith standard surface count

Details The Gromov invariant and the Donaldson-Smith standard surface count The Gromov invariant and the Donaldson-Smith standard surface count

2003-10-29

arxiv.org/abs/math/0310450v4

Geom. Topol. 8(2004) 565-610

Mathematics

Symplectic Geometry

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper14.abs.html

Scientific paper

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.

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