Geometry & Topology

The Gromov invariant and the Donaldson–Smith standard surface count

Michael Usher

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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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35]
Secondary: 57R17: Symplectic and contact topology

Pseudoholomorphic curves symplectic Lefschetz fibrations Gromov–Witten invariants


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.

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