  We introduce the notion of Artin motives and cohomological motives over a
scheme X. Given a cohomological motive M over X, we construct its punctual
weight zero part $\omega^0_X(M)$ as the universal Artin motive mapping to M. We
use this to define a motive 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 E_X.
  In the second half of the paper, we develop the application to locally
symmetric varieties. Specifically, let Y be a locally symmetric variety and
denote by p:W-->Z the projection of its reductive Borel-Serre compactification
W onto its Baily-Borel Satake compactification Z. We show that $Rp_*(\Q_W)$ is
naturally isomorphic to the Betti realization of the motive E_Z, where Z is
viewed as a scheme. In particular, the direct image of E_Z along the projection
of Z to Spec(C) gives a motive whose Betti realization is naturally isomorphic
to the cohomology of W.
