Datensatz

Zusammenfassung

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.