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The Gromov-Witten invariants of symplectic manifolds

Holger Spielberg
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Abstract

We study the fix point components of the big torus action on the moduli space of stable maps into a smooth projective toric variety, and apply Graber and Pandharipande's localization formula for the virtual fundamental class to obtain an explicit formula for the Gromov-Witten invariants of toric varieties. Using this formula we compute all genus-0 3-point invariants of the Fano manifold $¶(Ø_{¶^2}(2) \oplus 1)$, and we show for the (non-Fano) manifold $¶(Ø_{¶^2}(3) \oplus 1)$ that its quantum cohomology ring does not correspond to Batyrev's ring defined in \cite{bat93}.

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Holger Spielberg. 2000-06-21. The Gromov-Witten invariants of symplectic manifolds. https://arxiv.org/abs/math/0006156

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