The apparatus of motivic stable homotopy theory provides a notion of Euler characteristic for smooth projective varieties, valued in the Grothendieck-Witt ring of the base field. Previous work of the first author and recent work of D\'eglise-Jin-Khan establishes a "Gau\ss-Bonnet formula" relating this Euler characteristic to pushforwards of Euler classes in motivic cohomology theories. In this paper, we apply this formula to SL-oriented motivic cohomology theories to obtain explicit characterizations of this Euler characteristic. The main new input is a unicity result for pushforward maps in SL-oriented theories, identifying these maps concretely in examples of interest.

中文翻译：

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Motivic Gauss-Bonnet 公式

动机稳定同伦理论的装置为光滑射影变体提供了欧拉特征的概念，在基场的格罗腾迪克-维特环中有价值。第一作者之前的工作和 D\'eglise-Jin-Khan 的近期工作建立了一个“Gau\ss-Bonnet 公式”，将这个欧拉特征与推动上同调理论中欧拉类的推进相关联。在本文中，我们将该公式应用于面向 SL 的动机上同调理论，以获得该欧拉特征的明确表征。主要的新输入是面向 SL 的理论中推进映射的唯一性结果，在感兴趣的示例中具体识别这些映射。