Virtual classes and virtual motives of Quot schemes on threefolds

Abstract

For a simple, rigid vector bundle $F$ on a Calabi-Yau $3$-fold $Y$, we construct a symmetric obstruction theory on the Quot scheme ${\text{Quot}}_Y(F,n)$, and we solve the associated enumerative theory. We discuss the case of other $3$-folds. Exploiting the critical structure on ${\text{Quot}}_{{\mathbb{A}}^3}({\mathcal{O}}^r,n)$, we construct a virtual motive (in the sense of Behrend-Bryan-Szendrői) for ${\text{Quot}}_Y(F,n)$ for an arbitrary vector bundle $F$ on a smooth $3$-fold $Y$. We compute the associated motivic partition function. We obtain new examples of higher rank (motivic) Donaldson-Thomas invariants.

arXiv comment22 pages. Removed appendix. To appear in Adv. Math