We define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky’s category of motives. We prove that this motive can be written as a homotopy colimit of motives of smooth projective Quot schemes of torsion quotients of sums of line bundles on the curve. When working with rational coefficients, we prove that the motive of the stack of bundles lies in the localizing tensor subcategory generated by the motive of the curve, using Białynicki-Birula decompositions of these Quot schemes. We conjecture a formula for the motive of this stack, inspired by the work of Atiyah and Bott on the topology of the classifying space of the gauge group, and we prove this conjecture modulo a conjecture on the intersection theory of the Quot schemes.

中文翻译：

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关于曲线上的矢量束模块堆栈的VOEVODSKY动机

我们定义和研究Voevodsky动机类别中平稳投影曲线上固定秩和度的矢量束的模堆栈的动机。我们证明了该动机可以写成曲线上线束总和的扭转商的光滑投影Quot方案的动机的同伦同界。当使用有理系数时，我们使用这些Quot方案的Białynicki-Birula分解证明了束堆栈的动机在于由曲线的动机生成的局部张量子类别。我们根据Atiyah和Bott在规范组分类空间拓扑上的工作启发了一个用于该堆栈的动机的公式，并证明了该猜想是对Quot方案的相交理论的一个猜想。