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

LaTeX, 75 pages. Version as of November 1999

References in corpus (2)

• Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds
• Quantum Cohomology Rings of Toric Manifolds

Cited by in corpus (4)

• Localization in Gromov-Witten Theory and Orbifold Gromov-Witten Theory
• Comparing virtual fundamental classes: Gauge theoretical Gromov-Witten invariants for toric varieties
• Wall crossing for symplectic vortices and quantum cohomology
• A Combinatorial Case of the Abelian-Nonabelian Correspondence