Geometry & Topology, Vol. 8 (2004)
Paper no. 14, pages 565--610.
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.
Keywords.
Pseudoholomorphic curves, symplectic Lefschetz fibrations,
Gromov-Witten invariants
AMS subject classification.
Primary: 53D45.
Secondary: 57R17.
DOI: 10.2140/gt.2004.8.565
E-print: arXiv:math.SG/0310450
Submitted to GT on 18 December 2003.
Paper accepted 26 March 2004.
Paper published 31 March 2004.
Notes on file formats
Michael Usher
Department of Mathematics, MIT
Cambridge, MA 02139-4307, USA
Email: usher@math.mit.edu
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