Virtual intersections on the Quot-scheme and Vafa-Intriligator formulas

2005/05/31 by Alina Marian, Marian, Alina, Dragos Oprea +1 · 5 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.math/0505685

openalex publication_date 2005/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct a virtual fundamental class on the Quot scheme parametrizing quotients of a trivial bundle on a curve. We use the virtual localization formula to calculate virtual intersection numbers on Quot. As a consequence, we reprove the Vafa-Intriligator formula; our answer is valid even when the Quot scheme is badly behaved. More intersections are computed by the same method. Finally, we present an application to the non-vanishing of the Pontrjagin ring of the moduli space of bundles.

Cited by

• Intersection Theory of Hyperquot Schemes on curves
• A Borel--Weil--Bott theorem for Quot schemes on ℙ¹
• Curves in projective space and RSK
• Enumerating log rational curves on some toric varieties
• A short way of counting maps to hypersurfaces in Grassmannians

Related

• The virtual intersection theory of isotropic Quot Schemes
• Intersection Theory of Hyperquot Schemes on curves
• Virtual intersection theories
• Virtual cycles on projective completions and quantum Lefschetz formula
• Virtual Manifolds and Localization