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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 Eulercharacteristic. We extend the calculation to locally free sheaves on smooth$3$-folds, thus refining a special case of a recent Euler characteristiccalculation of Gholampour-Kool. We then extend Toda's higher rank DT/PTcorrespondence on Calabi-Yau $3$-folds to a local version centered at a fixedslope stable sheaf. This generalises (and refines) the local DT/PTcorrespondence around the cycle of a Cohen-Macaulay curve. Our approachclarifies the relation between Gholampour-Kool's functional equation for Quotschemes, and Toda's higher rank DT/PT correspondence.

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