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

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