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

2018/11/30 by Sjoerd Viktor Beentjes, Andrea T. Ricolfi · 15 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Combinatorics #Euler characteristic #Euler's formula #Mathematical analysis #Mathematics #Pure mathematics #Rank (graph theory) #Scheme (mathematics) #Sheaf #math.AG #msc:14N10 #msc:14N35

paper · pdf · open access · doi:10.4310/mrl.2021.v28.n4.a2

published in Mathematical Research Letters 28(4), 967-1032 (International Press of Boston) · v2. Minor changes and corrections following referee's comments, 40 pages. Accepted for publication in Math. Res. Lett

arxiv created 2020/02/03 · openalex publication_date 2021/01/01 · openalex created_date 2021/12/06 · arxiv updated 2022/03/15 · openalex updated_date 2026/07/20

Abstract

We show that the Quot scheme Quot_A³(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.

Citations

• Hilbert schemes of points for associative algebras
• MODULI OF REPRESENTATIONS OF FINITE DIMENSIONAL ALGEBRAS
• Hall algebras in the derived category and higher rank DT invariants
• The Geometry of Moduli Spaces of Sheaves
• A proof of the Donaldson-Thomas crepant resolution conjecture

Cited by

• Joyce's invariant and Virasoro Constraints for Quot schemes on curves