On the stack of 0-dimensional coherent sheaves: motivic aspects
Barbara Fantechi, Andrea T. Ricolfi
TL;DR
The paper addresses the problem of understanding motivic invariants of the stack $\mathscr{C}oh^n(X)$ of $0$-dimensional coherent sheaves by embedding it into the Grothendieck ring of stacks and connecting it to Quot schemes via Coh-to-Chow and Quot-to-Chow morphisms. It develops a framework of motivic decompositions using stratifications by support and singularity types, and demonstrates that punctual contributions depend only on local formal neighbourhoods, enabling explicit generating-function formulas in low dimensions. The main contributions include establishing motivic identities for $[\mathscr{C}oh^n(X)]$ and $[\mathrm{Quot}_X(\mathcal E,n)]$, constructing the punctual generating functions $\mathsf Z_{\sigma}(t)$ and $\mathsf Q_{r,\sigma}(t)$, and proving that punctual motives are controlled by local models like $\mathbb A^d$, all within the power-structure framework. This provides a foundation for explicit motivic computations in dimensions $\le 2$ and a roadmap for higher-dimensional extensions, with potential relevance to enumerative geometry and refined invariants.
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 $\mathscr{C}oh^n(X)$ of $0$-dimensional coherent sheaves of length $n$ on $X$. To do so, we review the construction of the support map $\mathscr{C}oh^n(X) \to \mathrm{Sym}^n(X)$ to the symmetric product and we prove that, for any closed point $p \in X$, the motive of the punctual stack $\mathscr{C}oh^n(X)_p$ 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 $\mathcal E \in \mathrm{Coh}(X)$.