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arXiv · 1811.09859

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

Sjoerd Viktor Beentjes·Andrea T. Ricolfi
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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.

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Sjoerd Viktor Beentjes, Andrea T. Ricolfi. 2020-02-03. Virtual counts on Quot schemes and the higher rank local DT/PT correspondence. https://doi.org/10.4310/mrl.2021.v28.n4.a2

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