Volume 8, issue 2 (2004)

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The Gromov invariant and the Donaldson–Smith standard surface count

Michael Usher

Geometry & Topology 8 (2004) 565–610

arXiv: math.SG/0310450


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.

Pseudoholomorphic curves, symplectic Lefschetz fibrations, Gromov–Witten invariants
Mathematical Subject Classification 2000
Primary: 53D45
Secondary: 57R17
Forward citations
Received: 18 December 2003
Accepted: 26 March 2004
Published: 31 March 2004
Proposed: Yasha Eliashberg
Seconded: Ronald Fintushel, Ronald Stern
Michael Usher
Department of Mathematics
Massachusetts 02139–4307