Geometry & Topology, Vol. 8 (2004) Paper no. 14, pages 565--610.

The Gromov invariant and the Donaldson-Smith standard surface count

Michael Usher

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.

Keywords. Pseudoholomorphic curves, symplectic Lefschetz fibrations, Gromov-Witten invariants

AMS subject classification. Primary: 53D45. Secondary: 57R17.

DOI: 10.2140/gt.2004.8.565

E-print: arXiv:math.SG/0310450

Submitted to GT on 18 December 2003. Paper accepted 26 March 2004. Paper published 31 March 2004.

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Michael Usher
Department of Mathematics, MIT
Cambridge, MA 02139-4307, USA

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