On the Voevodsky Motive of the Moduli Stack of Vector Bundles on a Curve

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 localizing tensor subcategory generated by the motive of the curve, using Białynicki-Birula decompositions of these Quot schemes. We conjecture a formula for the motive of this stack, inspired by the work of Atiyah and Bott on the topology of the classifying space of the gauge group, and we prove this conjecture modulo a conjecture on the intersection theory of the Quot schemes.

The Nisnevich Motive of an Algebraic Stack

“…Hence, it suffices to show that M (Y • ) ≃ M exh (X ) in DM eff (k, Z). The proof is exactly the same as [HL,Proposition A.7] using the atlas Y • → X .…”

A formula for the Voevodsky motive of the moduli stack of vector bundles on a curve

“…This paper is a continuation of our previous work [13] in which we define and study the motive M (Bun n,d ) ∈ DM(k, R) for any coefficient ring R (provided the characteristic of k is invertible in R in positive characteristic). More generally, we introduce the notion of an exhaustive stack and define motives of smooth exhaustive stacks by generalising a construction of Totaro for quotient stacks [16] (see [13,Definitions 2.15 and 2.17] for details). Using this definition, we describe the motive of Bun n,d as a homotopy colimit of motives of smooth projective Quot schemes by following a geometric argument for computing the ℓ-adic cohomology of this stack in [8].…”

“…Using this definition, we describe the motive of Bun n,d as a homotopy colimit of motives of smooth projective Quot schemes by following a geometric argument for computing the ℓ-adic cohomology of this stack in [8]. We use motivic Bia lynicki-Birula decompositions associated to G m -actions on these Quot schemes to further describe the motive of Bun n,d and based on these decompositions we conjecture the formula for the motive of Bun n,d appearing in Theorem 1.1 for a general coefficient ring R. In [13], we show this conjecture follows from a conjecture describing the interaction of the transition maps in the homotopy colimit for M (Bun n,d ) with these Bia lynicki-Birula decompositions; however, we were unable to prove this conjecture on the transition maps.…”

Infinitesimal deformations of some Quot schemes

“…Let Q := Q(E, d) be the Quot scheme parameterizing all torsion quotients of E of degree d. It is known that Q is a smooth projective variety of dimension rd. This Quot scheme has various moduli theoretic interpretations, see [BGL94], [BDW96], [BFP20], [HPL21], [BRa13], [OP21], which have led to extensive studies of it. Our aim here is to compute the cohomologies of the tangent bundle T Q , especially H 1 (Q, T Q ) that parametrizes the infinitesimal deformations of Q.…”