A Borel--Weil--Bott theorem for Quot schemes on $\mathbb{P}^1$

TL;DR

This work establishes a Borel–Weil–Bott–type description for the cohomology of tautological bundles on Quot schemes over $\mathbb{P}^1$, providing precise vanishing and stabilization results for Schur functors of these tautological complexes in large-degree regimes. The authors employ Strømme’s embedding into Grassmannians and a Koszul resolution, combined with a streamlined index theory for partitions and Horn inequalities, to reduce cohomology questions to Grassmannian cohomology and to isolate the first Koszul term. They prove new vanishing theorems for two consecutive line bundles and extend them to multiple insertions, yielding Ext-vanishing results and exceptional-collection structures that recover and extend Marian–Oprea–Sam results, including conjectures on exterior and symmetric powers. The results connect to quantum $K$-theory of Grassmannians and provide a framework for analyzing stabilizations of cohomology as $d$ grows, with potential implications for computing quantum invariants and the structure of derived categories on Quot schemes. Overall, the paper advances the understanding of tautological cohomology on Quot schemes on $\mathbb{P}^1$ and links these calculations to Grassmannian methods and quantum $K$-theory techniques.

Abstract

We study the cohomology groups of tautological bundles on Quot schemes over the projective line, which parametrize rank $r$ quotients of a vector bundle $V$ on $\mathbb{P}^1$. Our main result is an analogue of the Borel--Weil--Bott theorem for Quot schemes. As a corollary, we prove recent conjectures of Marian, Oprea, and Sam on the exterior and symmetric powers of tautological bundles.

Theorems & Definitions (54)

• Theorem 1.1
• Remark 1.2
• Corollary 1.3
• Remark 1.4
• Corollary 1.5
• Corollary 1.6
• Theorem 1.7
• Theorem 1.8
• Remark 1.9
• Theorem 1.10