Cosection localization and the Quot scheme $\mathrm{Quot}^{l}_{S}(\mathcal{E})$

Abstract

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 $\chi^{\mathrm{vir}}(\mathrm{Quot}^{l}_{S}(\mathcal{E}))=\chi^{\mathrm{vir}}(\mathrm{Quot}^{l}_{S}(\mathcal{O}^{\oplus r}))$.