Cosection localization and the Quot scheme ${\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}_S^l(\mathcal{E})$

Let $\mathcal{E}$ be a locally free sheaf of rank $r$ on a smooth projective surface $S$. The Quot scheme ${\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}_S^l(\mathcal{E})$ 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 ${\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}_S^l(\mathcal{E})$ is $(-1{)}^l$ times the fundamental class of the smooth subscheme ${\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}_C^l(\mathcal{E}{\mathrm{\mid }}_C)\subset {\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}_S^l(\mathcal{E})$. We then prove a structure theorem for virtual tautological integrals over ${\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}_S^l(\mathcal{E})$. From this we deduce, among other things, the equality of virtual Euler characteristics $\chi^{$\mathrm{v}\mathrm{i}\mathrm{r}$}($\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}$^{l}{S}($\mathcal{E}$))=\chi^{$\mathrm{v}\mathrm{i}\mathrm{r}$}($\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}$^{l}{S}($\mathcal{O}$^{\oplus r}))$.