The Gromov-Witten invariants of symplectic manifolds

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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 ¶(ض₂(2) ⊕ 1), and we show for the (non-Fano) manifold ¶(ض₂(3) ⊕ 1) that its quantum cohomology ring does not correspond to Batyrev's ring defined in bat93.