On the Voevodsky motive of the moduli stack of vector bundles on a curve

TL;DR

This paper defines and analyzes the motive of the moduli stack of vector bundles on a curve within Voevodsky's framework, expressing it as a homotopy colimit of motives of Quot schemes and exploring its structural properties.

Contribution

It introduces a new approach to understanding the motive of the moduli stack of vector bundles, relating it to Quot schemes and establishing foundational results in Voevodsky's category.

Findings

Motive expressed as homotopy colimit of Quot scheme motives

Motive lies in the subcategory generated by the curve's motive with rational coefficients

Conjecture for the motive formula of the stack, proven modulo intersection theory conjecture

Abstract

We define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky's category of motives. We prove that this motive can be written as a homotopy colimit of motives of smooth projective Quot schemes of torsion quotients of sums of line bundles on the curve. When working with rational coefficients, we prove that the motive of the stack of bundles lies in the localising tensor subcategory generated by the motive of the curve, using Bialynicki-Birula decompositions of these Quot schemes. We conjecture a formula for the motive of this stack, and we prove this conjecture modulo a conjecture on the intersection theory of the Quot schemes.