On the stack of 0-dimensional coherent sheaves: motivic aspects

Abstract

Let $X$ be a variety. In this survey, we study (decompositions of) the motivic class, in the Grothendieck ring of stacks, of the stack $\mathcal{C}o{h}^n(X)$ of $0$-dimensional coherent sheaves of length $n$ on $X$. To do so, we review the construction of the support map $\mathcal{C}o{h}^n(X)\to {\mathrm{Sym}}^n(X)$ to the symmetric product and we prove that, for any closed point $p\in X$, the motive of the punctual stack $\mathcal{C}o{h}^n(X{)}_p$ parametrising sheaves supported at $p$ only depends on a formal neighbourhood of $p$. We perform the same analysis for the Quot-to-Chow morphism ${\mathrm{Quot}}_X(\mathcal{E},n)\to {\mathrm{Sym}}^n(X)$, for a fixed sheaf $\mathcal{E}\in \mathrm{Coh}(X)$.

Final version. Minor revisions implemented, and added §2.2.1