# On the motive of the Quot scheme of finite quotients of a locally free sheaf

## TL;DR

This paper derives a formula for the motives of Quot schemes of finite quotients of a locally free sheaf on a smooth variety, extending previous results and providing explicit computations for curves and affine space.

## Contribution

It expresses the generating function of motives for Quot schemes in terms of the Grothendieck ring's power structure, extending known results to higher dimensions and relative motives.

## Key findings

- Derived a generating function formula for motives of Quot schemes.
- Extended previous curve-specific results to higher-dimensional varieties.
- Computed explicit motives for Quot schemes over affine space.

## Abstract

Let $X$ be a smooth variety, $E$ a locally free sheaf on $X$. We express the generating function of the motives $[\textrm{Quot}_X(E,n)]$ in terms of the power structure on the Grothendieck ring of varieties. This extends a recent result of Bagnarol, Fantechi and Perroni for curves, and a result of Gusein-Zade, Luengo and Melle-Hern\'{a}ndez for Hilbert schemes. We compute this generating function for curves and we express the relative motive $[\textrm{Quot}_{\mathbb A^d}(\mathscr{O}^{\oplus r}) \to \textrm{Sym}\, \mathbb A^d]$ as a plethystic exponential.