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The Gromov invariant and the Donaldson-Smith standard surface count. (English) Zbl 1055.53064

For any symplectic \(4\)-manifold \(X\), there exist symplectic Lefschetz pencils [see S. K. Donaldson, J. Differ. Geom. 53, No. 2, 205–236 (1999; Zbl 1040.53094)] which, after blow-up at the base points, yield symplectic Lefschetz fibrations. In S. Donaldson and I. Smith, [Topology 42, No. 4, 743–785 (2003; Zbl 1012.57040), arXiv:math.SG/0012067] the authors have introduced an invariant DS which counts pseudoholomorphic sections of a relative \(r\)-symmetric product constructed from the fibration. This should correspond to pseudoholomorphic \(r\)-multisections of the original fibration, raising the natural question of whether DS equals the Gromov-Witten invariant Gr introduced in C. H. Taubes [J. Differ. Geom. 44, No. 4, 818–893 (1996; Zbl 0883.57020)].
In [Serre-Taubes duality for pseudoholomorphic curves, Topology 42, No. 5, 931–979 (2003; Zbl 1030.57038), arXiv:math.SG/0106220], I. Smith has shown that (provided the fibration has high enough degree) DS satisfies a duality relation identical to that satisfied by Gr and proved in [C. H. Taubes, Seiberg-Witten and Gromov invariants for symplectic \(4\)-manifolds, (First International Press Lecture Series. 2. Somerville, MA: International Press) (2000; Zbl 0967.57001)], by relating Gr with the Seiberg-Witten invariants of \(X\).
This paper proves the conjecture that DS = Gr. The method of proof is to modify the almost complex structure \(J\) so that the curves \(C\) in the blown-up \(4\)-manifold corresponding to the count in DS are holomorphic, \(J\) is integrable near \(C\), the fibration map is pseudoholomorphic and \(J\) is generic. Special care should be taken with multiply covered tori.

MSC:

53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
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References:

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