# Virtual classes and virtual motives of Quot schemes on threefolds

## TL;DR

This paper constructs and analyzes virtual classes and motives for Quot schemes on threefolds, leading to new higher rank Donaldson-Thomas invariants and advancing enumerative geometry in Calabi-Yau contexts.

## Contribution

It introduces a symmetric obstruction theory for Quot schemes on threefolds and constructs virtual motives, expanding the framework for higher rank Donaldson-Thomas invariants.

## Key findings

- Constructed symmetric obstruction theory on Quot schemes
- Developed virtual motives for Quot schemes on threefolds
- Computed motivic partition functions and new invariants

## 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 $\textrm{Quot}_Y(F,n)$, and we solve the associated enumerative theory. We discuss the case of other $3$-folds. Exploiting the critical structure on $\textrm{Quot}_{\mathbb A^3}(\mathscr O^r,n)$, we construct a virtual motive (in the sense of Behrend-Bryan-Szendr\H{o}i) for $\textrm{Quot}_Y(F,n)$ 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.