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

TL;DR

This paper extends the Borel--Weil--Bott theorem to Quot schemes on the projective line, providing new insights into their cohomology and confirming recent conjectures about tautological bundles.

Contribution

It introduces an analogue of the Borel--Weil--Bott theorem for Quot schemes on , advancing understanding of their cohomological properties and resolving existing conjectures.

Findings

Established a Borel--Weil--Bott type theorem for Quot schemes on .

Proved conjectures on the structure of exterior and symmetric powers of tautological bundles.

Derived explicit cohomology calculations for tautological bundles on Quot schemes.

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.