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

TL;DR

This paper establishes a connection between shifted Yangians and the cohomology of Hyperquot schemes on curves, generalizing previous work and introducing skew-nested Quot schemes as key tools.

Contribution

It demonstrates that a shifted Yangian acts on Hyperquot scheme cohomology and constructs a natural basis using commuting Yangian operators, extending prior results from the case n=1.

Findings

Shifted Yangian acts on Hyperquot scheme cohomology.

A natural basis for cohomology is constructed via commuting Yangian operators.

Skew-nested Quot schemes are introduced as a new geometric tool.

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.