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 language | English |
---|---|
Pages (from-to) | 967-1032 |
Journal | Mathematical research letters |
Volume | 28 |
DOIs | |
Publication status | Published - 22 Nov 2021 |
Keywords / Materials (for Non-textual outputs)
- math.AG
- 14N35 (primary), 14N10 (secondary)