Let X be a smooth projective curve over a field of characteristic zero. We calculate the motivic class of the moduli stack of semistable Higgs bundles on X. We also calculate the motivic class of the moduli stack of vector bundles with connections by showing that it is equal to the class of the stack of semistable Higgs bundles of the same rank and degree zero. We follow the strategy of Mozgovoy and Schiffmann for counting Higgs bundles over finite fields. The main new ingredient is a motivic version of a theorem of Harder about Eisenstein series claiming that all vector bundles have approximately the same motivic class of Borel reductions as the degree of Borel reduction tends to $-\infty$.

中文翻译：

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希格斯丛模的动机类和有连接丛的模

设 X 是特征为零的域上的平滑投影曲线。我们计算 X 上半稳定希格斯丛的模栈的动机类。 我们还计算了带连接的向量丛的模栈的动机类，证明它等于相同的半稳定希格斯丛的堆叠类等级和度为零。我们遵循 Mozgovoy 和 Schiffmann 的策略来计算有限域上的希格斯丛。主要的新成分是关于爱森斯坦级数的 Harder 定理的动机版本，该定理声称所有向量丛都具有大致相同的 Borel 归约的动机类，因为 Borel 归约的程度趋于 $-\infty$。