The Gromov invariant and the Donaldson–Smith standard surface count

Usher, Michael

doi:10.2140/gt.2004.8.565

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The Gromov invariant and the Donaldson–Smith standard surface count

Michael Usher

Geometry & Topology 8 (2004) 565–610

DOI: 10.2140/gt.2004.8.565

arXiv: math.SG/0310450

Abstract

Simon Donaldson and Ivan Smith recently studied symplectic surfaces in symplectic 4–manifolds $X$ by introducing an invariant $D S$ 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 $D S$ satisfies a duality relation identical to that satisfied by the Gromov invariant $G r$ introduced by Clifford Taubes, which led Smith to conjecture that $D S = G r$ 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

Mathematical Subject Classification 2000

Primary: 53D45

Secondary: 57R17

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Publication

Received: 18 December 2003

Accepted: 26 March 2004

Published: 31 March 2004

Proposed: Yasha Eliashberg

Seconded: Ronald Fintushel, Ronald Stern

Authors

Michael Usher
Department of Mathematics MIT Cambridge Massachusetts 02139–4307 USA

Volume 8, issue 2 (2004)