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

Beentjes, Sjoerd Viktor; Ricolfi, Andrea T.

doi:10.4310/MRL.2021.v28.n4.a2

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

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Ricolfi, Andrea T.
Max Planck Institute for Mathematics, Max Planck Society;

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https://dx.doi.org/10.4310/MRL.2021.v28.n4.a2
(Publisher version)

https://doi.org/10.48550/arXiv.1811.09859
(Preprint)

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Citation

Beentjes, S. V., & Ricolfi, A. T. (2021). Virtual counts on Quot schemes and the higher rank local DT/PT correspondence. Mathematical Research Letters, 28(4), 967-1032. doi:10.4310/MRL.2021.v28.n4.a2.

Cite as: https://hdl.handle.net/21.11116/0000-0009-B8C9-5

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.