## Geometry & Topology

### 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.

#### Article information

Source
Geom. Topol., Volume 8, Number 2 (2004), 565-610.

Dates
Accepted: 26 March 2004
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513883408

Digital Object Identifier
doi:10.2140/gt.2004.8.565

Mathematical Reviews number (MathSciNet)
MR2057774

Zentralblatt MATH identifier
1055.53064

#### Citation

Usher, Michael. The Gromov invariant and the Donaldson–Smith standard surface count. Geom. Topol. 8 (2004), no. 2, 565--610. doi:10.2140/gt.2004.8.565. https://projecteuclid.org/euclid.gt/1513883408

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