We show that the Quot scheme $Q_L^n={Quot}_{{\mathbb{A}}^3}({\mathcal{I}}_L,n)$ parameterising length $n$ quotients of the ideal sheaf of a line in ${\mathbb{A}}^3$ is a global critical locus, and calculate the resulting motivic partition function (varying $n$), in the ring of relative motives over the configuration space of points in ${\mathbb{A}}^3$. As in the work of Behrend–Bryan–Szendrői, this enables us to define a virtual motive for the Quot scheme of $n$ points of the ideal sheaf ${\mathcal{I}}_C\subset {\mathcal{O}}_Y$, where $C\subset Y$ is a smooth curve embedded in a smooth 3-fold $Y$, and we compute the associated motivic partition function. The result fits into a motivic wall-crossing type formula, refining the relation between Behrend's virtual Euler characteristic of ${Quot}_Y({\mathcal{I}}_C,n)$ and of the symmetric product ${Sym}^{n}C$. Our ‘relative’ analysis leads to results and conjectures regarding the pushforward of the sheaf of vanishing cycles along the Hilbert–Chow map $Q_L^n\to {Sym}^n({\mathbb{A}}^3)$, and connections with cohomological Hall algebra representations.

中文翻译：

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本地动机 DT/PT 对应

我们证明了 Quot 方案 ${问}_{升}^n={报价}_{{\mathbb{一种}}^3}({\mathcal{一世}}_{升},n)$ 参数化长度 $n$ 线的理想层的商 ${\mathbb{一种}}^3$ 是全局临界轨迹，并计算所得的动机分配函数（变化 $n$)，在点的配置空间上的相对动机环中 ${\mathbb{一种}}^3$. 与 Behrend-Bryan-Szendrői 的工作一样，这使我们能够为 Quot 方案定义一个虚拟动机$n$ 理想捆的点 ${\mathcal{一世}}_C\subset {\mathcal{哦}}_{是}$， 在哪里 $C\subset 是$ 是嵌入平滑 3 倍的平滑曲线 $是$，我们计算相关的动机分配函数。结果符合动机穿墙式公式，细化了 Behrend 的虚拟 Euler 特征之间的关系${报价}_{是}({\mathcal{一世}}_C,n)$ 和对称乘积 ${符号}^{n}C$. 我们的“相对”分析得出了关于沿希尔伯特-周地图推进消失周期束的结果和猜想${问}_{升}^n\to {符号}^n({\mathbb{一种}}^3)$，以及与上同调霍尔代数表示的联系。