Geometry & Topology
- Geom. Topol.
- Volume 8, Number 2 (2004), 565-610.
The Gromov invariant and the Donaldson–Smith standard surface count
Simon Donaldson and Ivan Smith recently studied symplectic surfaces in symplectic 4–manifolds by introducing an invariant associated to any Lefschetz fibration on blowups of which counts holomorphic sections of a relative Hilbert scheme that is constructed from the fibration. Smith has shown that satisfies a duality relation identical to that satisfied by the Gromov invariant introduced by Clifford Taubes, which led Smith to conjecture that 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 in the blown-up 4–manifold that are made holomorphic by an almost complex structure which is integrable near and with respect to which the fibration is a pseudoholomorphic map.
Geom. Topol., Volume 8, Number 2 (2004), 565-610.
Received: 18 December 2003
Accepted: 26 March 2004
First available in Project Euclid: 21 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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