  Let $\mathcal{E}$ be a locally free sheaf of rank $r$ on a smooth projective
surface $S$. The Quot scheme $\mathrm{Quot}^{l}_{S}(\mathcal{E})$ of length $l$
coherent sheaf quotients of $\mathcal{E}$ is a natural higher rank
generalization of the Hilbert scheme of $l$ points of $S$. We study the virtual
intersection theory of this scheme. If $C\subset S$ is a smooth canonical
curve, we use cosection localization to show that the virtual fundamental class
of $\mathrm{Quot}^{l}_{S}(\mathcal{E})$ is $(-1)^{l}$ times the fundamental
class of the smooth subscheme
$\mathrm{Quot}^{l}_{C}(\mathcal{E}\vert_{C})\subset\mathrm{Quot}^{l}_{S}(\mathcal{E})$.
We then prove a structure theorem for virtual tautological integrals over
$\mathrm{Quot}^{l}_{S}(\mathcal{E})$. From this we deduce, among other things,
the equality of virtual Euler characteristics
$\chi^{\mathrm{vir}}(\mathrm{Quot}^{l}_{S}(\mathcal{E}))=\chi^{\mathrm{vir}}(\mathrm{Quot}^{l}_{S}(\mathcal{O}^{\oplus
r}))$.
