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The cohomology of Hyperquot schemes on curves via shifted Yangians in type A

Archi Kaushik

Abstract

Let $V$ be a vector bundle of rank $r$ on a smooth projective complex curve $C$. The Hyperquot scheme $\text{F}^{n}\text{Quot}\,(V)$ is the moduli space of length $n$ flags of rank $r$ sub-sheaves of $V$. This article has two main results: First, we show that a certain shifted Yangian of $\mathfrak{sl}_{n+1}$ acts on $H^{*}\left(\text{F}^{n}\text{Quot}\,(V)\right)$ by correspondences. Then, we define a family of $rn$ commuting Yangian operators which yields a natural basis for $H^{*}\left(\text{F}^{n}\text{Quot}\,(V)\right)$. This generalises the work arXiv:2307.13671 of Marian and Negut, who proved the above results in the case $n=1$. The new feature, which makes this generalisation possible, is the use of so called skew-nested Quot schemes. The rank $1$ versions of these spaces, skew-nested Hilbert schemes, have been recently introduced by Sergej Monavari in the context of refined DT theory of local curves arXiv:2506.14359. In the present article, skew-nested Quot schemes appear as correspondences associated with iterated commutators of Yangian elements.

The cohomology of Hyperquot schemes on curves via shifted Yangians in type A

Abstract

Let be a vector bundle of rank on a smooth projective complex curve . The Hyperquot scheme is the moduli space of length flags of rank sub-sheaves of . This article has two main results: First, we show that a certain shifted Yangian of acts on by correspondences. Then, we define a family of commuting Yangian operators which yields a natural basis for . This generalises the work arXiv:2307.13671 of Marian and Negut, who proved the above results in the case . The new feature, which makes this generalisation possible, is the use of so called skew-nested Quot schemes. The rank versions of these spaces, skew-nested Hilbert schemes, have been recently introduced by Sergej Monavari in the context of refined DT theory of local curves arXiv:2506.14359. In the present article, skew-nested Quot schemes appear as correspondences associated with iterated commutators of Yangian elements.
Paper Structure (40 sections, 66 theorems, 407 equations)

This paper contains 40 sections, 66 theorems, 407 equations.

Table of Contents

  1. Introduction
  2. Moduli spaces of sheaves and representation theory.
  3. Quot schemes of points on curves and shifted Yangians of $\mathfrak{sl}_{2}$
  4. Hyperquot schemes of points on curves and shifted Yangians of $\mathfrak{sl}_{n+1}$
  5. The commuting family of operators $a_{k}^{(v)}$ and a natural basis
  6. Related Works
  7. Organisation of the article
  8. Notations and Conventions
  9. Acknowledgements
  10. Geometry of Some Nested and Skew-nested Quot Schemes.
  11. The Hyperquot scheme ${\text{F}^{\text{n+1}}\text{Quot}_{{\vec{\text{d}}"_{k}}}\,(\text{V})}$
  12. Virtual projective bundles
  13. Equations defining some Hyperquot schemes
  14. The moduli space $\mathfrak{P}_{\vec{d}}$
  15. The moduli space $\mathfrak{Z}_{\vec{d}}$
  16. ...and 25 more sections

Key Result

Theorem 1.3

Theorems & Definitions (104)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Definition 2.1
  • ...and 94 more