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

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.

v2. Minor changes and corrections following referee's comments, 40 pages. Accepted for publication in Math. Res. Lett

References in corpus (5)

• Zero dimensional Donaldson-Thomas invariants of threefolds
• Hall algebras in the derived category and higher rank DT invariants
• The DT/PT correspondence for smooth curves
• Hilbert schemes of points for associative algebras
• A proof of the Donaldson-Thomas crepant resolution conjecture

Cited by in corpus (5)

• On the motive of the nested Quot scheme of points on a curve
• Tetrahedron instantons in Donaldson-Thomas theory
• Flags of sheaves, quivers and symmetric polynomials
• The motive of the Hilbert scheme of points in all dimensions
• Sur la lissité du schéma Quot ponctuel emboîté