Virtual counts on Quot schemes and the higher rank local DT/PT correspondence

Sjoerd Viktor Beentjes, Andrea T. Ricolfi

Research output: Working paper

Abstract

We show that the Quot scheme $\text{Quot}_{\mathbf{A}^3}(\mathcal{O}^r,n)$ admits a symmetric obstruction theory, and we compute its virtual Euler characteristic. We extend the calculation to locally free sheaves on smooth $3$-folds, thus refining a special case of a recent Euler characteristic calculation of Gholampour-Kool. We then extend Toda's higher rank DT/PT correspondence on Calabi-Yau $3$-folds to a local version centered at a fixed slope stable sheaf. This generalises (and refines) the local DT/PT correspondence around the cycle of a Cohen-Macaulay curve. Our approach clarifies the relation between Gholampour-Kool's functional equation for Quot schemes, and Toda's higher rank DT/PT correspondence.
Original languageEnglish
PublisherArXiv
Publication statusPublished - 24 Nov 2018

Keywords

  • math.AG
  • 14N35 (primary), 14N10 (secondary)

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