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

Abstract

Let $X$ be a smooth variety, $E$ a locally free sheaf on $X$. We express the generating function of the motives $[{\text{Quot}}_X(E,n)]$ 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-Hernández for Hilbert schemes. We compute this generating function for curves and we express the relative motive $[{\text{Quot}}_{{\mathbb{A}}^d}({\mathcal{O}}^{\oplus r})\to \text{Sym}{\mathbb{A}}^d]$ as a plethystic exponential.

arXiv comment17 pages. Minor revisions. Accepted for publication in J. Math. Pures Appl