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

Abstract

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

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