Mathematical Research Letters

Volume 28 (2021)

Number 4

Virtual counts on $\operatorname{Quot}$ schemes and the higher rank local DT/PT correspondence

Pages: 967 – 1032

DOI:  https://dx.doi.org/10.4310/MRL.2021.v28.n4.a2

Authors

Sjoerd V. Beentjes (School of Mathematics and Maxwell Institute, University of Edinburgh, Scotland, United Kingdom)

Andrea T. Ricolfi (Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy)

Abstract

We show that the $\operatorname{Quot}$ scheme $\operatorname{Quot}_{\mathbb{A}^3} (\mathscr{O}^r, n)$ 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 $\operatorname{Quot}$ schemes, and Toda’s higher rank DT/PT correspondence.

S.B. was supported by the ERC Starting Grant no. 337039 WallXBirGeom.

Received 9 September 2019

Accepted 27 January 2020

Published 22 November 2021