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The $L^2$-cohomology of a locally symmetric variety is known to have thetopological interpretation as the intersection homology of its Baily-BorelSatake compactification. In this article, we observe that even without theHermitian hypothesis, the $L^p$-cohomology of an arithmetic quotient, for $p$finite and sufficiently large, is isomorphic to the ordinary cohomology of itsreductive Borel-Serre compactification. We use this to generalize a theorem ofMumford concerning homogeneous vector bundles, their invariant Chern forms andthe canonical extensions of the bundles; here, though, we are referring tocanonical extensions to the reductive Borel-Serre compactification of anyarithmetic quotient. To achieve that, we give a systematic discussion of vectorbundles and Chern classes on stratified

On the reductive Borel-Serre compactification: $L^p$-cohomology of arithmetic groups (for large $p$)