The Gromov-Witten invariants of symplectic manifolds

2000/06/21 by Holger Spielberg, Spielberg, Holger · 2 citations
Mathematics · #14D (Primary) #14M25 #55N91 (Secondary) #58D #58F05 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.math/0006156

openalex publication_date 2000/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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) ⊕ 1), and we show for the (non-Fano) manifold ¶(Ø_¶²(3) ⊕ 1) that its quantum cohomology ring does not correspond to Batyrev's ring defined in \citebat93.

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