The Gromov-Witten invariants of symplectic manifolds

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 $\P(\O_{\P^2}(2) \oplus 1)$, and we show for the (non-Fano) manifold $\P(\O_{\P^2}(3) \oplus 1)$ that its quantum cohomology ring does not correspond to Batyrev's ring defined in \cite{bat93}.