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

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 $H^{\ast }\left({\text{F}}^n\text{Quot}(V)\right)$ by correspondences. Then, we define a family of $rn$ commuting Yangian operators which yields a natural basis for $H^{\ast }\left({\text{F}}^n\text{Quot}(V)\right)$. 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.