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arXiv · alg-geom/9702008

The Gromov Invariants of Ruan-Tian and Taubes

Eleny-Nicoleta Ionel·Thomas H. Parker
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Abstract

Taubes has recently defined Gromov invariants for symplectic four-manifolds and related them to the Seiberg-Witten invariants. Independently, Ruan and Tian defined symplectic invariants based on ideas of Witten. In this note, we show that Taubes' Gromov invariants are equal to certain combinations of Ruan-Tian invariants. This link allows us to generalize Taubes' invariants. For each closed symplectic four-manifold, we define a sequence of symplectic invariants $Gr_δ$, $δ=0,1,2,...$. The first of these, $Gr_0$, generates Taubes' invariants, which count embedded J-holomorphic curves. The new invariants $Gr_δ$ count immersed curves with $δ$ double points. In particular, these results give an independent proof that Taubes' invariants are well-defined. They also show that some of the Ruan-Tian symplectic invariants agree with the Seiberg-Witten invariants.

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BibTeXRIS

Eleny-Nicoleta Ionel, Thomas H. Parker. 1997-02-06. The Gromov Invariants of Ruan-Tian and Taubes. https://arxiv.org/abs/alg-geom/9702008

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