Relative Artin motives and the reductive Borel-Serre compactification of a locally symmetric variety

We introduce the notion of Artin motives and cohomological motives over a scheme X. Given a cohomological motive M over X, we consider the universal Artin motive mapping to M and denote it $\omega^{0}_{X}(M)$ . We use this to define a motive $\mathbb{E}_{X}$ over X which is an invariant of the singularities of X. The first half of the paper is devoted to the study of the functors $\omega^{0}_{X}$ and the computation of the motives $\mathbb{E}_{X}$ . In the second half of the paper, we develop their application to locally symmetric varieties. More specifically, let Γ\D be a locally symmetric variety and denote by $p:\overline{\Gamma\backslash D}^{\mathrm{rbs}}\to \overline{\Gamma\backslash D}^{\mathrm{bb}}$ the projection of its reductive Borel-Serre compactification to its Baily-Borel-Satake compactification. We show that $\mathrm{R}p_{*}\mathbb{Q}_{\overline{\Gamma\backslash D}^{\mathrm{rbs}}}$ is naturally isomorphic to the Betti realization of the motive $\mathbb{E}_{\overline{X}^{\mathrm{bb}}}$ , where $\overline{X}^{\mathrm{bb}}$ is the scheme such that $\overline{X}^{\mathrm{bb}}(\mathbb{C})=\overline{\Gamma \backslash D}^{\mathrm{bb}}$ . In particular, the direct image of $\mathbb{E}_{\overline{X}^{\mathrm{bb}}}$ along the projection of $\overline{X}^{\mathrm{bb}}$ to Spec(ℂ) gives a motive whose Betti realization is naturally isomorphic to the cohomology of $\overline{ \Gamma\backslash D}^{\mathrm{rbs}}