Let X be a smooth variety, E a locally free sheaf on X. We express the generating function of the motives $[{\mathrm{Quot}}_X(E,n)]$ in terms of the power structure on the Grothendieck ring of varieties. This extends a recent result of Bagnarol, Fantechi and Perroni for curves, and a result of Gusein-Zade, Luengo and Melle-Hernández for Hilbert schemes. We compute this generating function for curves and we express the relative motive $[{\mathrm{Quot}}_{{\mathbb{A}}^d}({\mathcal{O}}^{\oplus r})\to \mathrm{Sym}{\mathbb{A}}^d]$ as a plethystic exponential.

中文翻译：

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关于局部自由捆的有限商Quot方案的动机

令X为平滑变种，让E为X上的局部自由捆。我们表达动机的产生功能$[{\mathrm{报价单}}_X（Ë，ñ）]$就品种的Grothendieck环的力量结构而言。这扩展了Bagnarol，Fantechi和Perroni的最近曲线结果，以及Gusein-Zade，Luengo和Melle-Hernández的希尔伯特方案的结果。我们计算曲线的生成函数，并表达相对动机$[{\mathrm{报价单}}_{{\mathbb{一种}}^d}（{\mathcal{Ø}}^{\oplus \mathrm{[R}}）\to \mathrm{象征}{\mathbb{一种}}^d]$ 作为体积指数。